It's a common design flaw in B/W games of the era. You'll need to replace the connector and possibly the pins if the scorching is bad enough.

Seen much worse....u sld be ok just replacing the connectors and add some leds to playfield gi and backbox. Just a few can take alot of strain off the db

Replace them -- both the header and connector. If you don't have a desoldering gun, invest in one. You will use it again and again with board repairs (blown transistors, fuse clips, and headers), and they pay for themselves the first time you don't pull a trace or screw up a solder pad.

It's common in a lot of Classic Bally, Classic Stern, Game Plan, Sega, Data East games too...not just B/W. Interesting that it was very common among many manufacturers for so long. One of the simplest circuits in the game is the one with the most problems (+5V watchdog circuit excluded).

To the OP, as mentioned earlier, replace both the header pins and connectors/housings. You'll probably notice the playfield lights get brighter after you do. But be extremely careful with the header pins when removing them, even when using a desoldering iron. The plated through holes are susceptible to damage.

Excellent suggestions throughout the thread. Especially about procuring a desoldering tool. The Hacco one is a pleasure to use. Not cheap but again as another said, worth every fine when you don't screw up traces on a board.

Double checked the schematic and it looks like J120 and J121 are 11 pin connectors (0.156"): (Great Plains) CS156-11-LR. The Keying pin part number is 15-04-0219.
I would get both size trifurcon connectors: 0.156" 18-20AWG 08-52-0113 (Great Plains) and 0.156" 22-26AWG 08-50-0185 (Great Plains). Maybe get 100 a piece and get that 15% discount? Going by the picture I believe the 22-26AWG contacts would fit nice.
Of course you'll need a crimper and I found this one to work a dream: https://www.amazon.com/gp/product/B007JLN93S (HT-225D)

Lastly you'll most likely want to replace the headers and the part number is 26-48-1245 (Great Plains)

As lyonsden mentioned you may want to invest in a desoldering tool. You either can get the real deal Hakko FR-300 or the knock off Aoyue 474A++ https://www.amazon.com/Aoyue-Digital-Desoldering-Station-Built/dp/B00ABJ4AEC/ref=sr_1_4)

Get the connectors and headers from pinballlife, also get the punchdown tool from there.

Don't use the obviously-not-up-for-the-job IDC connector housings. Get the crimp-n-slide-in type and use trifurcon pins. That hardware is woefully under-engineered for the purpose.

IMO!

I agree with this!

Don't use IDC connectors. Molex are the same price or cheaper.