First of all, that's a ton of schools! Good luck!!! Second of all, why aren't your positions liquid?
I thought you could always buy or sell anything, and that there was no need to wait for expiration. Most importantly, since you seem to be very familiar with the inner workings of high finance, have you ever heard of the float? How is it that you have access to the options and futures market, but not to a credit card? With a credit card, you could submit today, and not actually need the cash for up to a month after a statement is generated, which can be as much as a month after today, depending on when your statement cycle ends.
FWIW, the $2K you are going to drop on your primaries is nothing compared to the $5K you are going to drop on secondaries. If you are spending your last $7k on med school applications, maybe you shouldn't be in the options and futures market in the first place.
Disadvantage is relative. It's early, so submitting today vs. a month from now is not going to determine your fate. OTOH, getting in a month earlier will give you an indication as to how things are going for you a month earlier. In your situation that might have some value to you.
Bottom line, if you can't access your cash and don't have access to a credit card, it is what it is, so what difference does it make? Otherwise, there is no reason not to submit as soon as they are ready. Another thought might be to push the primaries out now and stagger the secondaries. You are going to have a difficult time submitting 55 secondaries all at once, so you will have a built in delay in any event. On top of that, you might not receiving all 55 secondaries, and they very likely won't all come at once. Just something to think about.