Hey dude, yup, you're gonna have to take that stepper unit apart. You can do it without desoldering anything.
First, the stepper coils come out, and you can let them dangle by their wires. Then, the wiper fingers - the "spider" - comes off. Before you do that, you should reset the unit to zero and take a picture to remind yourself which way up it goes.
From there, you can remove the circuit board from the back of the unit, and let it dangle by its wires.
Finally, IIRC, this stepper unit has a switch to tell it when it's at its first position, and another switch to tell it when it's at its last. Take pictures of both so you remember which one goes where and facing which way, then remove them carefully - the screws like to fall out and into the depths of the machine, so maybe put a light-coloured blanket or sheet on top of the mechanism panel. Sometimes a switch will fall apart, sometimes it won't, this is why you take pictures first.
Now you have no wires attached to the mechanical part of the stepper unit, and you can take it off for inspection.
The stepper unit is removed via the screws at its base, that go up into the playfield. Take it off and take it away somewhere where you can sit down and work on it.
You'll need to take apart everything on that stepper unit. Take pictures beforehand. Each pivoting arm, the central spindle - it's all got to come off, because it'll all be filthy. Be careful with that central mainspring, it'll whip round and bite your finger if you give it half a chance. Likewise, when you remove the C-clips, they'll enjoy pinging off to find the darkest corner of the room, underneath the heaviest piece of furniture you own.
I use isopropyl alcohol and a microfibre rag to clean the spindles, and the inside of the shaft where the central spindle moves. It'll all be covered in black gunk. Use the rag, the alcohol, and if necessary, fine sandpaper or emery cloth. Make sure that everything moves freely. If you use sandpaper on anything, remember to wipe it down with your alcohol rag afterwards to get rid of any sanding dust. That goes for switch contacts too. Clean absolutely everything, with one exception - there's a nut and screw which affect the landing position of the step-up solenoid's armature. Leave that screw and nut alone for now.
Some people lubricate the pivot points with a light machine oil - I personally tend to avoid lubrication, because I find in the long term it causes more problems than it solves. Up to you. Remember, only steel on steel needs lubricating - steel on nylon, no lube. Steel on brass, no lube.
Now, once the mechanics are back together and moving smoothly, you'll need to inspect the contacts on the spider, and on the circuit board. These will probably need sanding a bit. Make them nice and shiny.
You have to ensure that the spider comes to rest precisely on the contacts it's supposed to come to rest at. You have two adjustment points for this - firstly, there's a little screw and nut that affects where the armature of the step-up coil comes to rest. Secondly, the circuit board has some leeway in how it can be positioned on the stepper unit. Fiddle with one at a time until you've got it centered and correct.
If you like, PBResource.com sells a conductive contact disc lubricant. Use very little. Good luck.