Thank you very much! I took a look at the Google search and found out a little more on those peach borers you were all talking about -it's one nasty little critter!
I just ran outside to take a closer look at the spots that are weeping. I couldn't get up close to the upper (and larger) wound, but the smaller one did have a bit of cobweb-like stuff. However, when I dug down around the roots - they look good! No sign of cobwebs down there at all, in fact, I noticed a lot of small black ants around the base of the tree.
I sprayed "Spectracide" tree patching spray on the wounds I could reach (there's actually one large, one small and one that just might be a natural crack in the bark). I'll do the largest one tomorrow afternoon, when I can get help putting up the larger ladder.
Anyway, I guess it's time to start learning about pruning and arbor (tree) management.
Thanks again all and best wishes in your gardens too!
P.S. If I come across any info while searching on peach trees, or weeping in general, did y'all want me to post a link here for it?