I was with Aela and I think she died. We were in a fight with four mammoths and two giants. Things got hairy (no pun intended) and I called down a storm and I am pretty sure she was killed. I haven't seen her for quite some time now. I couldn't find her body though. Anyway, I can't get another traveling companion now. Everytime I try, they tell me that it looks like I already am with someone now.

Have you tried fast travelling?
If your companion is still alive that will usually bring them back to your side.

She's alive if you can't get someone new to join. Try to backtrack to that mammoth fight. Wait an hour, fast travel, area transition and so on. Something will hopefully get her to pop up.

Definitely try fast travelling somewhere. If she's still alive and just lost somewhere, she should head back to Whiterun. Lydia got stuck in a never-ending fight on a mountain during my first playthrough. Instead of killing her opponent, she'd walk away when he yielded and he'd attack, only to get the beatdown again. I left her to it and she never rejoined me during my trip to the Greybeards. She wound up waiting for me at the house in Whiterun.

Have you checked the Companions place in Whiterun? Usually when they get lost they return to where you got them.