If I made an offer for the full asking price on a short sale, can the bank now change the price?

In a short sale, the bank is not actually the other party--the seller--to the transaction: the other party is still the homeowner, since the bank has not  yet foreclosed. The homewner technically does not even need bank approval to sell short; anyone can sell their home for essentially any amount. The homeowner needs bank approval of  the sale--and selling price--though if he's going to be able to avoid liability for the remaining balance of of his mortgage; if all the various lenders do not agree to the transaction, then if the home is underwater and the seller sells short, the seller will still be liable for any amount left on the mortgage not paid off by the sale.

In the case  you describe, it is likely that one of the banks has decided that, based on the appraisal, the seller  should be able to get more for the home and therefore pay off more of the mortgage. The bank is probably saying to the seller that it will not accept the short sale as payment in full of all mortgage liability unless he gets more money. The bank is allowed to do this: since it is not the owner of the home and not a party to the sale--but has the seller contractually obligated to it, under one or another mortgages--it can state that it must get a certain amount or it will not let the seller  off the hook for the unpaid balance of the loan. The seller in turn has little or no reason to go through with the sale, unless he knows that by doing so, he will clear his indebtedness in full.

So the short answer is, banks do not have to approve short sales at all and may set any amount as the threshhold or minimum price to accept one. If there is a signed home sale contract, the seller would have to go through with the  sale, even if he's underwater; but until there is an executed contract binding the seller, the banks can change what they would consider an acceptable amount, and almost any seller will go along with what they say. Note that as a general matter, a full-asking price offer *never* guarantees you'll get the home--sellers may reject full asking price offers for any reason, including that they do not have bank approval for a short sale.