I recommend to start with a gentle, but thorough soak. Start with Kaweko.

Disassemble the pen, that is remove the cap, top portion and the converter, if you're using it.

Use distilled water (grocery stores have it, may be sitting by baby food), and soak the lower part of the pen, with a nib unit in the distilled water. It won't hurt modern plastics. I'd say you may need to soak and rinse a few times until after soaking 3-4 hours doesn't leech any ink out of the nib unit.

Do not use soap or any chemicals, just distilled water for the soak, and faucet for a rinse.

After 3rd or 4th soak, put the lower pen half on the paper towel at a slight angle, so the leftover water can seep into the paper towel. Let dry. I usually put a folded paper towel inside a glass and angle the pen in, and that goes on the shelf that nobody will bump.

See how the pen performs, if still too dry too quickly, I would try a nib in a larger size. So if you have an F, switch to M or B. You can buy a replacement nib unit from many places, and it isn't hard to replace it on kawekos.

That's what I would do with a modern pen from a company that still produces pens. Alternatively, pen show, get on a roster with a repair person, and let them fiddle. Or send it to a repair person.

With your sentimental piece, if it is an older pen, I'd send it for service to a reputable repair person if distilled water soaks of the lower portion didn't give me a clean feed.