I would bet it's food related. My Jake came home on iams at 8 weeks and we switched him immediately. There are many people here who have much more knowledge in the food area..a lot of bullies here love the Kiddy pools..I doubt 100% it's that! How convenient the previous owners "forgot" her records! They can get a release to their vet and then you can get them..somehow I wouldn't be surprised they don't follow through for you! She is adorable and what a relief she has found you! Get her on a good food...find a good vet and be prepared for bully love!!
Also stress can make her skin problems worse..and the change of a new home is stressful for them..also I'm sure others will also advise you that when you make a food change..go slowly!