That's kinda crappy that you have to work full time to help your siblings through med school, which thereby hurts your chances at getting into a school yourself. Why not loans? I hope they return the favor when you're in med school!
Your sGPA is not competitive at MD schools (average is a 3.6), and it's very hard to determine your chances without an MCAT score. Have you taken any practice tests so you can have a rough estimate of what your score will be? It will be an uphill battle for MD school, although not impossible. A good MCAT score along with the upward trend in your GPA will give you a good shot at some schools. I suggest buying the MSAR to figure out a school list. I think you'd be fine for DO school GPA-wise.
As for LORs, most schools require at least 2 letters from science faculty, and you only have 1. You should definitely look into another letter from a professor unless you know the schools you are applying to do not require 2 science letters. Typically a research letter does not substitute for these letters, but every school has different requirements. Letters from MDs you shadow are generally not highly regarded, so instead of that second MD letter, get one from a science professor you had.
A July 26 MCAT means you won't be complete until the end of August, which can be considered late for someone with a low GPA. Again, not impossible, but not the best scenario. Don't move your test date up if you're not ready. You also don't need to wait for your LORs to be done to submit - if you are going to apply this cycle, submit as soon as your personal statement and work/activities are finished so that you can get verified. Your LORs are not needed until you are receiving and submitting secondaries. (This is advice for AMCAS, I didn't apply DO so I don't know AACOMAS rules).
If you really want to increase your chances at MD schools, consider applying next year when you've had a chance to increase your GPA even more and get a second science letter.