Since the burn damage may have damaged the logicboard, I'd inspect that before you spend too much money on the phone. If you find out the problem is isolated to the connector and the board survived, then there should be no problem with just replacing the connector.
If the motherboard didn't survive, you're probably better off getting a new phone. New boards are very expensive and used boards are a gamble in terms of usability; this isn't a problem with a new phone.
While it is possible to do a bare replacement with solder, it's a difficult thing to do. The connector is inexpensive so it really isn't worth the effort (or gamble) to reuse the old assembly. Even then, this is an advanced soldering job.
iPhone 4 Verizon Dock Connector Replacement
iPhone 4 Verizon Logic Board Replacement
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