You can definitely do a combined MA/PhD. Some schools advertise the option and others don’t. If there’s a particular school you’re interested in that has both programs but doesn’t advertise a combined option, you can contact the department to see if it’s possible. I started my search by looking at the programs that advertised a combined MA/PhD option on EdFind.
If you know you want a PhD, there are definitely advantages to going straight through: you only have to apply once, you only have to move once, you’ll be finished sooner (as in sooner in your lifetime), it might streamline the process somewhat (i.e., 6 years instead of 7 total between MA/CFY/PhD), you’ll be working on research with the same mentor for longer, etc. Also, if you apply to a combined program you might be considered for funding reserved for PhD students, and schools usually have more money to fund PhD than masters students.
I would recommend attending the ASHA Convention if you can (it’s in November in Denver this year, and you can get your registration waived if you're accepted to volunteer). I did that last year and it was incredibly helpful. I talked to programs at the grad school fair and set up meetings with students at one of my prospective schools. Also check out the resources on ASHA’s website.
I'm starting an MA/PhD program in the fall. I also want to teach, and I plan to get my CCCs during the program. If this is the path you want, go for it! The field definitely needs more PhDs.