Don't get the lead free solder. Sure, it works, but it has a higher melting point and therefore much harder to work with. I've been soldering for +40 years and I still hate it. Some good good 'ol Kester 60/40 or 63/37 (i.e. solder with lead) with rosin is so much easier to work with.

Use some solder wick to remove the original solder because it's likely lead free. And then apply the solder mentioned above (and also mentioned in another comment). You'll be fine w/o rosin flux. No harm in using tho if you want too.

Never, ever, use plumber's flux! It's very corrosive and is not intended for electronics.

Excellent advice slochar. Reiterating his comments, the solder connection should provide the electrical connection, not the mechanical connection. Follow that rule, and it will be good for ages.