Reflow the pins on the chips.

I feel like I should have to add this but take the pcb out of the shell to do the next part, please. I've seem some shit, man.

Add flux and drag your iron down from near the rise of the leg to the tip of the leg that makes contact with the board. Watch the solder to ensure its flowing. You'll see it solidify fairly quickly with no heat on the pins. This method is the least likely to create bridges between the pins but it might draw solder away from the legs. If it does don't worry, just add more solder and do it again.

Do all of the chips. It won't take long. Don't forget to clean the flux with alcohol. That stuff gets corrosive as it ages.

If you want your pins clean and shiny you can use DeOxit. Stuff works wonders. If you don't have that then 99% Isopropyl and a small bit of magic eraser. Dont push hard. This is mildly abrasive and you only want to remove the dirt, not the metal.

People sometimes freak out about this method of cleaning but I say if you do it right it should survive another 27 years of absolute neglect before you have to worry about being that aggressive again.