It's sort of a silly problem that I guess I'm missing something since I'm new into the biking world, but anyways, I have an older Peugeot PE10 that was working fine until I had the rear tire and tube replaced. After the replacement, I noticed there was a lot of drag coming from the rear tire. I've checked the brakes, clean the bearings , hub, even thought it may have been a broken freewheel which I took off and cleaned out in preparation for a replacement. Just for kicks, I put the wheel back on without the freehub and I noticed that the problem was actually whenever I tighten the quick release skewer securely, it locked up the wheel and created the drag. If I leave it loose enough to spin freely, the wheel wobbles quite much, so much so that it easily rubs against the brakes even though I've released the tension off the brake cable during all my cleaning/repair attempts. I've double checked if everything was put back properly a few times over but the problem persist. Anyone have any idea whats going on?

The problem has to of course be with the hub, either locknuts not tightened or damage to cups/cones. As you've overhauled the hub it can't be my first guess - a broken able. You need to make sure the locknuts are properly tightened against the cones, and if so take the hub apart again and inspect the surfaces. Also, do not rise the balls.

like CNY-Bikeman I first suspected a broken axle, but hat would b obvious to you since you greased the hub.

So there are a few possibilities, such as the bearings being tight, and the added compression form the closed QR making the difference.

OTOH, I think you might look elsewhere. The photo shows a bent hanger, it's possible or even likely that the dropouts aren't parallel. That means the closed skewer flexes the axle into an arc binding the bearing. Straighten the hanger which needs to be done anyway, and while at it bring the dropouts to parallel and try the wheel again.