I have an Axim x3 with the same issue. Dell won't repair. They will replace with a refurbished model for $199.00. The entire motherboard has to be replaced because the connector is part of the motherboard. I have searched for online places to replace it for me and none are willing. One suggestion is to find a broken model (with an intact motherboard) and replace it yourself. However, I have not had success with this either. Some sites sell motherboards for the Axim x5, but I guess the x3 is not yet available as a separate part. So either wait for that to occur or buy a new unit. If it is under warranty you should contact Dell (be preapred to wait on the phone forever).
I have a dell axim x3 got a new mugen 1100mHz battery and started using it but the battery gets discharged without using at all and I screwed up the power socket the connector pins and am not able to install any prog into it. please help me what to do!!!