I'm going to guess that the school with the summer start is Monmouth
I applied to only a couple of schools last year (wanted to take a gap year to gain volunteer experience, but figured I'd try a couple anyway), both being in Jersey. When I applied to Stockton and Monmouth, I still had to take A&P Speech/Hearing (I was taking them in the Spring). Ultimately I didn't get in, but that was due to subpar gre scores, really. I had no problems applying when I didn't complete the pre-reqs yet.
I can't really help you with the pre-req situation for MU, unless you're willing to take on one more course. MU used to allow students to take the pre-reqs after being admitted into the program (I believe it was in first one or two years after establishing the program), but now they make it mandatory to have everything completed prior to matriculation. Still, I would contact them just in case. Right now their pre-req courses are online, so if possible I'd take advantage of that. I shadowed under an SLP at a hospital who was a supervisor for MU's program and they're planning on making the courses either hybrid or fully in person. WARNING: Phonetics with Prof. B is awful, if that's the course you need. She doesn't really understand how to work online classes.
I'm applying to fours schools in Jersey as well; hoping I can stay in state lol. I'm also applying to schools out of state too, despite not wanting to leave here, because I had a taste of how competitive the programs are around here. Last year MU had 300 applicants and only took about 25. Crazy. I'm not even attempting to apply to Montclair lol.
Good luck! If you have any questions, feel free to ask!