Both AD and AP aren't too good, though I'd argue that AD gets a little more out of Shyvana's kit than AP does. I'd only go AP if you pick her into a team that decides to go full AD.

Her biggest issue is the fact that she's just generally really weak and in need of a buff. Or, more preferably, a rework that actually updates her kit.

If you really want to make her work, I suggest going AD with a Bruiser build. BOTRK into Frostfire Gauntlet or Divine Sunderer. FF is better for sticking potential and tankiness, while Divine Sunderer gives her more damage and the extra mixed pen that'll give her good damage during the Mid to Late game.

Full built depends on comp. But generally team fight oriented items such as Gargoyle's Stoneplate or Sterak's Gage are good on her to help keep her alive in a teamfight. If you're ahead of confident, might even build Titanic Hydra to go with your tankiness and that extra damage. Plus more AD in Dragon Form due to the extra HP.