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Sorry for the late reply on this. Another thing you might consider doing is to trade letters with a friend who is also applying so that your drafts have an obviously different "voice". (In other words, you write one of their drafts, and they write one of yours.) I've done this a couple of times with good results.

It is pretty common for referees to ask students to draft LORs themselves, and admissions committees almost surely know that, so I wouldn't be too worried. That said, I sympathize with your having to do all three yourself-- I am having to draft one on my own and worry that it will be too similar to my SOP and writing sample, as I feel that I have a somewhat distinct writing style (at least, when multiple pieces are laid out next to each other).
I would recommend a few things. Firstly, look up LOR samples. Donald Asher's book, Graduate Admissions Essays, has some samples; and I'm sure there are others online. Try using samples to draft letters that have different formats from each other. Secondly, be aware of elements you fall back on in your own writing. Are there words you use often? Sentence structures you lean towards? Be aware of these and avoid them as much as possible. You don't have to pretend to be a totally different person -- I think pretending that you didn't write something when you did would be the worst option here (dishonesty is never a good look on an applicant) -- but you can make the LORs less obviously written by yourself.

Might want to try something along the lines of:
"Thank you very much for the acceptance to your program!  I am very excited by the idea of attending (insert university) this (insert application term).  I will be taking this offer into serious consideration when making my final decision.  Thank you again for taking the time and providing me with the opportunity to share my passion in (insert field) with your university."
Universities like knowing you have multiple offers.  It makes them feel like they made the right decision, in my opinion.

I use something like what you wrote, plus "I will consider this offer most seriously." 
I also emailed that to all the individual faculty I interviewed with + "thank you for the interview and the positive words you must have put in."
Don't know if this is appropriate but I feel comfortable sending these out, so that' what I did.

Hi! I'm super excited because I just got my first response from a school--an acceptance from one of my top programs. I feel like I should respond to the acceptance email with something like "Thank you so much! I am honored to have been accepted to this program and am incredibly grateful for this scholarship..." but I'm still waiting on seven other schools, so I can't accept their offer yet. I'm terrible with small social stuff like this, so any thoughts on how to proceed would be highly appreciated! Thanks.

This is actually a favorite pastime of mine. Great topic.
 
"0a/0w/2r 11 The Night is Dark and Full of Terrors"

Too late is when it arrives too late to be considered. So 3 days is probably fine. Many committees will not meet until January so you might have lots of time still.
If a school has a ton of applications, they might do some work in December to collate/sort the files and prepare packages for the committee to review first. If a school doesn't then all materials might go directly to the committee. As long as your materials get to the profs before they start making decisions, it will be fine. 
Some schools may throw out incomplete applications, but I don't think many schools will reject an applicant for an incomplete item out of their control (i.e. if it's missing a SOP then that might be rejected but not if it's missing a LOR). 
That said, what you should do now is to contact the schools waiting for this letter and let them know that your prof has told you he will be a little bit late in submitting the letter. Make sure you contact both the graduate school / university-level people as well as the specific department you are applying to. Don't make excuses for your prof or complain about it or anything. Just state what has happened so that they know to expect the letter a little late.