Do you have the pickguard off? If so are the wires still soldered from the pickguard to the tremolo claw and to the output jack? You have to have it grounded to the trem claw and the hot has to go to the output jack for this to work right.
Also, do you use soldering flux? This greatly helps. You brush some paste flux on the intended solder point after roughing up the site slightly, then place the wires at the site to be soldered to. You want the wires at the bottom of the solder pile. The solder pile should be kept to a minimum. brush on some paste flux as you got to speed up the soldering process. You should heat up the soldering site and wires adequately so that when you touch the solder wire to the wires, the wires draw the solder into the strands; rather than pileing the solder on top of cool wires. The soldering flux will greatly help to get a good solder joint. I always use solder that contains LEAD, and I use thin solder wire, about one half a mm in diameter, maybe a little thicker. I get it at Radio Shack, one of the only places carrying leaded solder.
At the switch you want your solder sites to be independent of other solder sites, with none of the solder joints touching, even slightly, other solder joints; except at the central solder joint where all the grounds are soldered together in one solder joint.
Finished solder joints should be shiny and not be huge piles of solder heaped on top of other piles. Use only enough solder to get a good joint and start with a good and hot solder site before applying solder to the wires and site. Don't heat the solder wire and expect it to drop onto the wires and solder site and hold the wire there. Heat the site first and draw in the solder when you apply the solder to the site, not to the soldering iron tip. Watch that your soldering iron doesn't melt the insulation off surrounding wires, get them out of the way even if you have to tie them off. Don't use a soldering gun, use a soldering pencil or a soldering iron. I have a twenty five watt one and a bigger fourty watt one for quickly heating up soldering sites, then I use the small one to get in there with precision.
Try to use these principles even when you solder in the old pickups.
When you put on the Neovins did you ever put the pickguard on the guitar and tune up the strings and listen to how it sounded in place? All the wires have to be connected and you won't know how it will sound if you don't pluck a string over the pickup.
Hope this gives you some ideas. I've had to resolder pickups myself to get the joints right and the connections right. Sometimes it can be difficult, especially when the replacement pickups are not wired like the old ones.
I personally would try not soldering to the switch box and the note in the instructions about soldering the silver shielding to a separate ground from the black ones would encourage me to solder them to a different pot or something other than the common ground where all the black wires are grounded.
Also I would not hesitate to email GFS for further assistance, even though it might be weak assistance, if any. Seymour Duncan, no affln., has a technical support guy you can call on the phone, just as an example of superior customer service provided by a premium pickup maker.
I would agree that you need to work on your soldering skills so that you can get good solder joints. Using paste soldering flux will GREATLY simplify getting a good solder joint for a beginner or even an old timer having difficulty. If the solder wants to roll off of the solder site, you need soldering flux brushed on there to burn off the impurities and make the solder joint a place where the solder can adhere without needing to roll off.