I always patch, but in the wet and the cold I'll swap the tube for my spare and patch in the comfort of a dry warm place. I don't believe you can properly apply a patch if the tube is wet. I've had none of the glueless patches I've applied last more than a couple of months, so I only use glued patches now.
In the case of the OP, I'd replace the tube as getting the old glue off is not worth the effort.