It may be that the connector is loose on the PCB. Re-wetting the solder joints may fix it, but the surface mount contacts are quite small and usually beyond the skills of most. If you are not very confident soldering small surface mount components I would not suggest attempting it. Permanent damage to the PCB is possible. There are specialized micro soldering stations for surface mount components. I have used them in my previous profession, but I do not own one. I can do limited surface mount soldering with a regular soldering station and a very pointed tip.
A usual pencil iron is too large on the end. It will require a very pointed tip. You may also need to use a lighted magnifier. I would not use lead free solder. There is no reason not to use 60/40 solder in order to use the lowest possible heat. Lead free solder is used in manufacturing to meet RoHS compliance.
The solder contacts are so small they probably need no more heat than is necessary to melt the solder on the soldering iron tip.
Your best option may be to get it repaired by Uniden or maybe Gerry Oliver at G & G Communications.
