You could find a "copy shop" like Kinko's. At least as of mid-2005 there was one not far from Goodings Market. Past Downtown Disney, out Hotel Plaza Blvd, left onto Rte. 535 (Goodings was straight ahead), straight through next light and turn into a plaza half a block further on the right. There (hopefully still) is a nice inexpensive buffet restaurant there too for those not on the dining plan.
If you are well versed emptying the Flip into your own PC and also burning DVD's, it shouldn't take that much pay by the minute practice to burn a DVD using one of their computers. You will need to bring any supporting or driver software the Flip came with.
I would be more confident burning two DVD copies before erasing/reusing the built in memory.