Ok, those pins may look solid, but if that jack can move that much, I'm betting the solder joints on the back of that board are in really bad shape. @RobinsonNi and others would be able to confirm that. It may be as 'simple' as resoldering those joints as previously suggested. At this point, I would take the board out to take a closer look at this.
Looking at the packet capture, I don't see any major issues, besides the fact that your PC is only getting some, not all responses back. This usually means it's a layer 1 issue.