I don't recall any specific injury, but I think it happened while deadlifting/squatting a little over a week ago. Can I keep running on it and just avoid weights? It doesn't really hurt when I run, but it just feels really tight and hurts a little when I stretch it, kind of on the inside back part of my middle thigh. I haven't run in 5 days, and have been trying to avoid stretching or loading it, and it feels like it's just getting stiffer. So, hence why I'm wondering if running will help keep it loose. Or will it just slow down recovery?

When I self diagnosed a pulled hamstring four years ago, I definitely couldn't run on it. I found a recovery plan online and followed it religiously. Don't think I ran for four weeks but I didn't reinjure it.

You probably just strained it. Do nothing for three weeks and try something light. When I pulled a hamstring I did nothing for 4-5 weeks except rehab.

You would know if you really pulled it. Painful to walk, stand and sit(especially on the crapper).
When it happened to me, a bruise about the size of a softball emerged - black as black could be.
It took me 9 months to recover in my twenties.

Hamstring strains suck. You may have a mild one. Typically you have to wait at least several days after you no longer have symptoms before you start easing back into activity. Plenty of rest in the mean time.