I overdid my Single-leg Romanian Deadlifts this weekend. I was in a hurry and wasn't performing the exercise with proper control, which has led to me overstretching them a bit. I'm back to the point where I can walk without limping, but they do still hurt some. Is it safe to do running and my upper-body workout (push-ups, tricep dips, let-me-ins, and let-me-ups) for now?
I would recommend doing something, to second Alex here, for active recovery.
I would avoid running and do some skip rope instead and after you properly warmed up, you can do your other exercise. The body is a unit and you will be using your hamstrings with the other exercises anyway indirectly.
You should be really sure that it is really soreness that you feel, if you somehow injured the muscle itself - it would feel different, I would not do something until the pain is gone.
Just a personal preference, on days where I was overdoing it, I used ice-massage after the training. Simply put a water cup (small paper cones) fill it with water and let it freeze. Use some Oil on your skin before you massage the body part until the ice melted.
Some people recommend stretching, but I personally have never felt any benefit out of it rather feeling worse afterwards. ..
Bummer about Delayed Onset Muscle Soreness (DOMS) in your hamstrings: it's one of the more annoying places to get it and makes it tough to do pretty much everything. Barbell good mornings pretty much guarantee me DOMS.
I think you're fine, but start with a walk, then get to a warmup, and see what's comfortable. Usually once the blood is flowing it doesn't really hurt that much (technical: the pain is attenuated), although it will come back pretty much as soon as you stop.
I've had DOMS bad enough that I couldn't run, and other times where it was okay. It's really variable. The good news is that every few hours the pain should be a bit less after it peaks, so your luck is going up.
Upper Body Lifts
I wouldn't think this is a problem at all, and honestly it's a pretty normal way to keep lifting despite injury (sprains, strains, and in your case, a nasty case of DOMS).
I often hear people say that it's completely safe to exercise with DOMS, and while that may be technically true I think it's a little narrow. In my own experience, the problem comes from the pain causing you to use bad form. With your hamstrings, as an example, during a squat you might shift your weight to your toes and round your back, in order to keep the pull off of your hamstrings. It's not conscious by any means, but rather a natural result of your body trying to keep you from using something that is in pain.
Is it safe to do my upper-body workout?
Yes. The most that your hamstrings will be doing during these will be acting as a stabilizer.
Is it safe to do running?
Maybe. Start off slow, and make sure to warm up thoroughly (maybe more than usual) to avoid stretching your hamstrings while cold. While working up to a running pace, pay attention to how you feel. If they hurt (more than just being sore) you're better off stopping or slowing down to a pace that doesn't make them hurt.
As an aside, you may also benefit from some active recovery directed at your hamstrings to get more blood flow to them.