I had the same problem, haven't kept a master logbook in years, and stopped keeping a red crew log about 5 years ago, just printed up my line from the company computers at the end of every month to keep a running total on my flight time.
The problem is that doesn't account for night or IMC time or actual instrument approaches shot. At 9,000 hours, I really don't give a flying rat fart, but obviously they do, so I had to guestimate it as best I could in my master log with one line entry for each year.
They said I was successful in my application, but ranked your class entry by how well you did in the sim. Unfortunately for me, ATA went out of business at almost exactly the same time, so a bunch of current and qualified 757/767 people interviewed right after me and, of course, did better in the sim, being rated in the aircraft and currently flying it. So I never was placed for class, and my application packet purged after a year.
As far as the American Medical? It's a freaking joke. Always has been, anywhere in the world. Not even nearly as thorough as a standard yearly doctor checkup for a non-pilot, much less screening for someone with our responsibilities.
However, the JCAB medical was a non-event for me, if a little lengthy. The only thing I have is borderline-high cholesterol levels because I like my cream in my coffee every morning and my steaks and ribs on a regular basis. I'm 5' 11" and 170 pounds and run and work out 5 days a week, so the BMI wasn't a problem, but my cholesterol never goes below 200 and stays closer to 230. Even so, I still passed. If you're in halfway decent shape, don't sweat the medical. If your BMI is bad, well...
Question for the others who are there: If they're overstaffed, why another recruiting drive?