New caliper seals.
I often wondered how the self-adjusting feature works until I found out. The seal between the piston and the caliper is designed to flex under normal operation, as the piston moves in and out. If the pads have worn and the seal flexes past a certain point, the piston will slip through the seal a little bit and take up the slack and when you let go the seal returns and the piston is in a new position.
I think that if the seals get old or haven’t moved for a long time they stick to the seal so this doesn’t happen, or it takes more flex for it to happen, the result of which is that the bite point gets closer to the bars. Given that your brakes are old this is probably it I reckon.
New seals will render the brakes good as new. The flexing seal is basically the only moving part in these – so if you replace it you’ll be good to go.