Is it normal to get still elbows after riding a while, or should I try to move my saddle up so my arms are more bent?

I would suggest looking at the tilt of your seat. I have found that if the front of the sadlle is to low you will continually be pushing yourself back on the seat and fight to keep from sliding forward again. Even if the seat looks level try bringing the nose up a bit. The saddle fore and aft position is more for fine tuning your leg psoition. A stem change would be for the reach that you are talking about. I would take a quick look here https://www.peterwhitecycles.com/fitting.htm as well as doing a search in the forums for bike fitting.

I'm not sure if my stiff elbows went away becuase I rode more or because I adjusted my seat (height, tilt, fore-aft) until my butt quit hurting after in the saddle for only 30 mins!

Yours elbows should not be anywhere close to locked in any position. Especially if you're racing - bent elbows are good for absorbing bumps from other riders.