EdPhDHopeful

Personally, I would not spend upwards of $60,000 on getting another Master's degree unless I suddenly decided to change career paths and was confident that this expenditure would payoff in the future (I think the estimated income for new HGSE grads is like $57,000??), but that's me.
Being brutally honest, your current GRE scores are a bit of a concern... and since you're still quite young, I would recommend studying for them again (if you have the financial means to do so). I am totally the same when it comes to testing (my first round of scores were not the best) but I took a few weeks to study and ended up increasing my scores quite a bit (above Harvard's averages in V & AWA). Yes, GREs are not everything and do not showcase an applicant's full academic capabilities, but they are still somewhat important and the "I never do well on standardized tests" reasoning doesn't stand as well when you have a Master's IMO. 
I would recommend taking a year or two to teach or do work in communities that you are passionate about and to experience the real world of supporting students. Yes, you have quite an extensive resume for a straight-out-of-undergrad/3+1/co-term applicant. However, from my own experience, I think that being a full-time teacher/educational advocate (I'm assuming that your Student Teacher experience was an in-person teaching requirement/internship for your Master's in Teaching?) truly exposes you to the intricacies of educational systems and systemic disparities that hinder the success of students. Hopefully such experiences will also allow for personal/career development that would better serve you in deciding if you truly want to pursue another graduate degree. Likewise, because you do have strong references, I would simply stay in touch and continue to try working with said professors.
However, as a college counselor, I would still encourage you to apply if you feel that this is the best path for you at the current moment! We never know what admissions committees are looking for as overall institutional goals constantly change from year to year so you may very well get in! ^^ 

Personally, I would not spend upwards of $60,000 on getting another Master's degree unless I suddenly decided to change career paths and was confident that this expenditure would payoff in the future (I think the estimated income for new HGSE grads is like $57,000??), but that's me.
Being brutally honest, your current GRE scores are a bit of a concern... and since you're still quite young, I would recommend studying for them again (if you have the financial means to do so). I am totally the same when it comes to testing (my first round of scores were not the best) but I took a few weeks to study and ended up increasing my scores quite a bit (above Harvard's averages in V & AWA). Yes, GREs are not everything and do not showcase an applicant's full academic capabilities, but they are still somewhat important and the "I never do well on standardized tests" reasoning doesn't stand as well when you have a Master's IMO. 
I would recommend taking a year or two to teach or do work in communities that you are passionate about and to experience the real world of supporting students. Yes, you have quite an extensive resume for a straight-out-of-undergrad/3+1/co-term applicant. However, from my own experience, I think that being a full-time teacher/educational advocate (I'm assuming that your Student Teacher experience was an in-person teaching requirement/internship for your Master's in Teaching?) truly exposes you to the intricacies of educational systems and systemic disparities that hinder the success of students. Hopefully such experiences will also allow for personal/career development that would better serve you in deciding if you truly want to pursue another graduate degree. Likewise, because you do have strong references, I would simply stay in touch and continue to try working with said professors.
However, as a college counselor, I would still encourage you to apply if you feel that this is the best path for you at the current moment! We never know what admissions committees are looking for as overall institutional goals constantly change from year to year so you may very well get in! ^^ 

@NativeEd I submitted my application about two weeks ago and did not see my scores under the bolded "Standardized Test Scores" section. As such, I contacted GSAS and they informed me that scores do not automatically get matched with applications immediately after submission. It takes a day or two for the application system to put the two together. Seeing "GRE General Official (date of exam)" under "Standardized Test Scores" will mean that GSAS has received and matched your GRE scores. 
It should look like this (I blurred out my references and GRE exam dates):

I think it would be dependent on what kind of programs you are looking at. Applying for a Master's program straight from undergrad would most definitely be easier than applying to a doctoral program. Check out HGSE's "Who Studies at HGSE?" page. You'll see that there were students without any work experience in both the Ph.D. cohort and Master's cohort from the 2016-2017 entering class. But the catch is also that while the age range for the Master's cohort was 20-61, the age range for the Ph.D. cohort was 23-37, which means that the youngest member might have gone into a two-year Master's program straight from undergrad before applying OR they could have just taken a bit longer with their undergrad career OR they could have completed uni and gotten a few years of work experience prior to their acceptance... there are a multitude of plausible explanations for possible correlations between the age range and years of work experience, which is to say that it's difficult for anyone to give you a direct answer detailing your level of admissibility in regards to post-undergrad work experience.  
Personally, I think it's totally fine to apply this year if you have the funds, are totally ready with the appropriate application requirements, and feel that it is the most appropriate step for your future. Likewise, we, as applicants, never know what HGSE is looking for. Institutional goals and benchmarks are changed regularly so you may very well get into the program of your choice. BUT, as you are set to graduate this year, I would definitely have a backup plan ready just in case admission results do not go well... aka you probably should start preparing yourself for job/career fairs after you submit your application if you choose to apply for admissions this year.    

Personally, I would not spend upwards of $60,000 on getting another Master's degree unless I suddenly decided to change career paths and was confident that this expenditure would payoff in the future (I think the estimated income for new HGSE grads is like $57,000??), but that's me.
Being brutally honest, your current GRE scores are a bit of a concern... and since you're still quite young, I would recommend studying for them again (if you have the financial means to do so). I am totally the same when it comes to testing (my first round of scores were not the best) but I took a few weeks to study and ended up increasing my scores quite a bit (above Harvard's averages in V & AWA). Yes, GREs are not everything and do not showcase an applicant's full academic capabilities, but they are still somewhat important and the "I never do well on standardized tests" reasoning doesn't stand as well when you have a Master's IMO. 
I would recommend taking a year or two to teach or do work in communities that you are passionate about and to experience the real world of supporting students. Yes, you have quite an extensive resume for a straight-out-of-undergrad/3+1/co-term applicant. However, from my own experience, I think that being a full-time teacher/educational advocate (I'm assuming that your Student Teacher experience was an in-person teaching requirement/internship for your Master's in Teaching?) truly exposes you to the intricacies of educational systems and systemic disparities that hinder the success of students. Hopefully such experiences will also allow for personal/career development that would better serve you in deciding if you truly want to pursue another graduate degree. Likewise, because you do have strong references, I would simply stay in touch and continue to try working with said professors.
However, as a college counselor, I would still encourage you to apply if you feel that this is the best path for you at the current moment! We never know what admissions committees are looking for as overall institutional goals constantly change from year to year so you may very well get in! ^^