I don't know about the requirements for the specific schools you listed nor about the relative merits of switching your major. However, math wise, I'm a little bit confused about the chronology you have listed. Do you need to retake algebra? If so, why wait until summer to take both algebra and calculus? (I'm on a quarter system, you may be on a semester system). Why not take algebra next semester, then calculus over the summer? Or consider brushing up on your algebra by yourself, then take calculus next semester. I feel like waiting from now all the way until summer to take the rest of your math is not ideal, as you lose math skills pretty quickly without regular work on them. I also feel like taking concurrent math classes would be harder than necessary, especially if you need those skills from algebra for your calculus class (and now, for your pre-cal class.
I don't know what your current study plan for pre-cal looks like, but I had good success with studying for a small amount of time each day - an hour or so. I usually would look over my notes from the previous day, work several problems related to the topic covered, and then if there was homework due weekly, I'd work on some problems from the homework each day (rather than doing all of the homework on one day).
Also, you may not need pre-cal as a specific PT school prereq, but you may need it as a prereq for the prereqs. My school may be unique, but I needed a quarter of pre-cal for chemistry, and 2 quarters of pre-cal to start the physics series (which is a rule they imposed after students with algebra only historically did poorly/failed).