Thanks to all for your input. The common suggestion appears to be that I pull the windows, clean out old caulk, and recaulk.
Ozz, the caulking used on my windows was the black sealant. The problem with it is that it appears to expand in the heat and then harden where it then flakes away, leaving air gaps where water can penetrate. I am looking for something that will remain pliable, not shrink and not bleed.
I like the idea that Trailer Trash 2 suggests. It seems to fit just the need (no oils to bleed, will not shrink, and is permanent).
Now I wish I had a larger shed to store monty in! Would make it a bunch easier to do this job inside instead of out in the weather.