Gatorskin tires should be good, if there isn't something in them. I periodically probe the bottom of every hole in the tires for glass, especially if I mount an unknown used tire.
Some people orient the valve with either the tire label or tire pressure markings. I usually just pull tire and tube off together. Do a quick check for the hole with them together, then if I don't find it, remember where the valve was and pull it apart.z
Make sure your valves are 100% straight. Especially with Schrader valves. Before inflating, you can squeeze the tire and roll it in a circle to get the valve straight.
Depending on how bad the wheel is, you may be able to true it. I have wondered if a tire too far out of true causes some lateral stress on the tube, and can cause it to leak in unexpected ways (around a patch?).
New parts never hurt, and can make the riding experience much more pleasant.