Hi! I am a first year doc student in School Psych and can hopefully answer your questions.
First of all, deep breaths! Second of all, it's not too late to apply, especially if you are looking at applying for EdS programs instead of PhD. The first thing you need to do is decide which you want to apply to. Let's start with this: what is your career goal? If it is to work as a School Psych in a school, EdS is probably sufficient. If you want to do something fancy like be a professor or work in a hospital, you'll need a PhD. That is a minimum of a 5-year commitment.
EdS: Master's+. This takes about 2 years of school (a Master's degree) + thesis/project + a year of internship. For this degree there is no expectation to do research, you do not need to contact profs before applying. You do not need to do any research at all (although you can if you want). The deadline for these programs is anywhere from January to March. These typically do not fund, although they can. At my school (very good about funding) only one of the EdS students is funded, and she really lucked out in finding it.
PhD: Doctorate, of course. About 4 years coursework (on average) + 1 year internship + dissertation. This qualifies you for more things, most of them more technical. Totally not worth it if you just want to be a SP in schools and that is it forever. To apply for this you need research experience, you need to choose a school based on their research interest and mentoring styles. Most deadlines are ~December 1-15. The (arguably) top school in the field only takes applications until November 15. You should find a school that funds you for a doctorate. Please, please do not pay out of pocket for 5 years.
In my program Doc students and EdS students take almost the exact same classes until the 3rd year, when doc students carry on with school and EdS ones leave for internship.
The next trouble is accreditation. Make sure ALL programs (EdS/PhD) that you apply to are NASP approved. It's a bonus at the doctoral level if it is NASP and APA approved. Always go with NASP first (NASP does not actually accredit doc programs... but they are considered NASP accredited if their accompanying EdS programs are). This will save you a lot of misery when you go to get licensed or if you are licensed/working and switch states.
Definitely read this info sheet about SP from NASP. And is the School Psych thread if you are interested in joining us! You can also PM me if you want.