I think the question is, is the finish dulled from wear, or from accumulated and oxidized oils (from skin)?

I have had great success cleaning both painted and stained wood with a damp cloth and liquid dishwater detergent, followed by a clean damp cloth rinse/wipe -- removing sometimes a great deal of soil (including waxy buildup from the dog's coat) without any damage to the finishes.

I would try that first -- if that restores the shine, great, it was just dirt; if not, you will have to try something to restore the finish.

I agree....a lot is likely just dirt...
What is the finish? lacquer reacts with skin oils to form a dingy film....a good household cleaner will remove it.

The hand oils have combined with the lacquer based finish to form gunk. When cleaned if the gunk, it will also remove the damaged finish.

You need to think about your long term plans. Refinishing the cabinets is extremely labor intensive and so will be as expensive as replacing.

Old English, Oil soaps, and Orange Glop are the worst possible substances to use on cabinets. The oils in them ruin the finish. The only way to remove the damage this things cause is to sand it all to bare wood. Unfortunately, if any residue of the stuff gets into the crevices if where the joins are, the refinish will fail.

I disagree that the only way to fix the "gunk" is to sand to bare wood. Use a cleaner to remove the gunk and then wipe the area with an quick drying oil finish....like Min Wax antique oil finish.
You can also remove all the old finish with lacquer thinner....because factory build cupboards are almost always finished with lacquer.
Oh and old English is mineral oil and won't compromise a lacquer finish.

Oils are what combine with the lacquer to make the gunk in the first place. Only old cabinets are finished with actual lacquer. It’s too moisture sensitive for good wear. And it’s brittle and tends to flake on wood.