This is getting really bad, especially in my knees. I can't leg press enough to really help my legs because my knees can't take as much as the muscle can. Not to mention after cleaning out one of the rooms in our apartment yesterday, much of my joints are so swollen and sore, especially my left elbow and shoulder, it's left me nauseous. Does anyone have any methods of relieving arthritis pain and swelling?

Glucosamine/chondroitin works for me.  I have the same problem as you - my muscles get stronger faster than my joints.  The only solution I've found is to lighten up on the weight and do more reps.  It does improve over time.

I've been dealing with bad knees for years.  YOu can do some things to help, but bottom line - getting old sux.  You are going to have to make adjustments in your workouts/expectations. 
As far as the things that help: Glucosamine/chondroitin (already mentioned) is very good.  Though some folks say that chondroitin (the more expensive part of that combination) is so poorly metabolized as to be not very valuable in pill form.  So I just take the glucosamine and didn't notice any drop off after stopping the chondroitin.  I also take MSM and believe it helps AT LEAST as much as the glucosamine.
If you're doing the same exercises every workouut, try changing them up so as not to be stressing your knees in the same way all the time. You may have to give up some exercises completely. Ultimately, I had to give up every form of leg press / squat.  But I can still go heavy on the knee extensions (as long as I don't lock 'em out all the way at the top).  This does not allow me to put on the mass I'd like, but I've maintained what I had and am nicely cut. Good luck.

Well I tried suggesting something some time ago.
Did you ever try that? Though I know the idea wasn't very appealing.