Yeah, is that volume pot lug grounded at all?
Don't think so. Should it be? Where?
You have effectively 3 wires. The outer braid is the green and shield (grey) together. As both of them are grounded with every wiring diagram applicable to 2-conductor pickups they are simply not needed to be separated out into 2 wires. Then you have the inner hot wire which goes to the switch. In this case that is the white wire with your pickups.
The reason why the blob is positioned like that is that the tab is bent back to the pot back and soldered there in many guitars. The other method is to take the cut end of the cap leg and bridge from tab to pot.
You will need a ground wire connecting between the pots.
With regard to the pot back soldering - see if you can get the solder onto the wires without them on the pots. Once you can get them tinned then they might stick better to the solder on the pots.
Warm the pot then put the wire in the puddle of solder, and try and use the tip to 'hoosh' the excess over the top of the wire.
In these sort of cases you need to trace the signal. Check between the pickup wires and the switch ground to see you are getting the approx right K reading for that pickup. Select the pickup with the switch and check that the signal is making it through the switch - there are electrical and mechanical issues to check.
Also check your new ground wire isn't connecting on the cap leg.