I think your schedule is fine. I wouldn't start looking until May and look for a college student. Many classes are MWF or TT. Maybe you can find someone with TT only classes. Post at the colleges on the bulletin boards.

OP, I think you'll be able to find someone. We have someone who works 3 days a week, the schedule has changed from time to time, and the 3 days have never been subsequent (we do M/W/F now, and did Tues/Thurs/Fri before). She wants PT work because she goes to school 2 days a week (but is in her 30s--she's not a college kid per se). We started at $17/hour with some benefits (paid vacation, paid federal holidays) because that seemed to be what it took to get someone reliable for PT work. I think you are likely looking too early--I personally would not be looking 6 months out for a PT job. We hired our current nanny about a month or two before she started, so maybe looking over the summer would be more realistic as grad students and other people who may only want PT work will have a better idea of their schedule. We also considered a nanny split but it seemed like there would be just a ton of variables and in the end I couldn't find another family that wanted only 2 days a week--though it's not out of the question.
I would also suggest that you expand your search beyond Care.com. We tried that several times and I found it to be like online dating--people sounded great in their profiles but in person it was really hard to find a good fit. We found several good candidates through DCUM and a neighborhood listserv. Good luck!

Op, what city are you in and what hourly rate can you afford for a good person?

You can post an ad looking for a share family on this site, or on your local list serves. Also check with neighbors and friends with similarly aged children who might be interested.