I have this fluffy white gunk growing all over the stem of my one yellow tomato, and now its spreading to my other tomatoes. It's only on the stems, but its all over the the plants from top to bottom (meaning it's not concentrated near the roots). It seems to appear quickly where there is a wound where I've pruned the plant. Most of the leaves on the plant have yellowed and are dying. I've cut off most of the dying leaves, but if I cut off all the diseased leaves, the plant would have no leaves left and I don't think it would be very happy.

I've tried looking up what it is, but there doesn't seem to be a good online resource out there for tomato diseases with good big pictures. I can't post a picture right now because my camera is dead, and I can't find a good picture online that looks similar. It's been really hot where I live and in the last couple weeks there's been a lot of rain, but I've had this problem for a couple months. Should I kill the plants before the disease spreads to my herbs? I have a container garden on my balcony.