No...it's not really considered late. It's late when your file is complete @ about the same time as August MCATers. Their files are generally On Hold until their scores arrive in October. IMHO, you're starting to be "late" when September hits.
Definitely, have someone look over your PS. Make sure it's glossy!
Your instincts are correct. Contact the department to find out when your profs might be in their offices. Explain your situation...the fact that you needs LoRs ASAP. The dept should be able to help out. Science profs understand the drill. You're not the first premed who needs a LoR. To be on the safe side, I asked 3 profs for letters - just in case one wasn't able to come through. Profs have other letters to write, classes to teach, grant apps to be completed. I had my timeline.
Additional letter writers...volunteer cooridinator/director, a physician you shadowed, church leader, etc. may enhance your app.