How would I attach a decently sized part onto the player without the player constantly tipping over?

For example, I want to add wings to my character, but the character is really heavy with the wings welded to it. How would I stop this?

You can try set Locked,Anchored and Collide false.

You could instead, try constantly changing the parts position to be near the player. If you really do want to weld the parts, I would suggest trying something with body velocity.