I had been counseled about being late to work and falling asleep at work. I was late one day after that and I asked a fellow employee to say that I was on time. She told on me. They confronted me about it a few days later. I had decided that I was not going to lie on my time card but when my Sgt screamed at me I told him I was there at 7:30 (a lie). I resigned from the position because the union representative told me that would be better than being fired. I have applied for another position at the base as they never had any problem with my work. I have since changed doctors and have been told that the medication and the amount of it was the cause of my sleeping and not being able to wake up. Can they hold this against me for this other job.

I assume you had a civilian, union job.

Ask your union, but my guess is, while they may or may not be able to hold the problems with the medication against you, they certainly can hold the dishonesty against you.