With a 96% match rate for PGY-1 & 2 positions overall, it is still very, very competitive. Also, most of the FMG's admitted to anesthesiology programs are going to armpit programs that can't attract enough domestic graduates. LSU Shreveport, Arkansas, OU, etc. account for the lion's share of FMG's and aren't likely to be able to attract better grads any time soon.
So back to the original poster's questions, yes you can still get into an anesthesiology program if you don't mind living in a hell hole, working like cheap labor for a private group, not getting any didactic education, and/or not getting enough experience in all aspects of anesthesiology. Even then, the match rate is still high and only the very worst programs have had available positions for scrambling.
A few years ago, a colleague of mine used LSU Shreveport as his practice interview. When they realized that he was a domestic graduate, he was immediately offered a position outside the match and a salary of $40K if he would sign on the spot. The money was tempting for him, but the inconsistencies of the interviewers and PD (interviewers told him there were no hearts or transplants available and he had to be farmed out to Corpus Christi or Baton Rouch for pedi, Texas Hearts for a single month of hearts, etc., while PD told him they had all of the above in Shreveport) and the fact that is was, well, Shreveport ("all of your patients have to be checked against the incest list") convinced him to leave halfway through his interviews.
Nail your next boards, do the externships, get letters from the professors at your externships, and you will get into a good program. It is well worth the effort.