Kitchen cabinets and Restor-A-Finish

Overall, I love my kitchen. Someday when we have extra money (ok, so like in 15 years when the home equity loan for the addition is paid off?), we'll re-do the countertops and put in a tile floor. But we love our cabinets, which are a beautiful cherry:

However, the cabinets that get direct sunlight have started to fade--quite a bit--in the 20+ years since they were installed:
Here you can see the parts covered by the doors/drawers that are NOT sun-faded:
So i was complaining about them on an email list and my dear friend Suzee suggested a product called Restor-A-Finish. She swears by it, saying that it saved her a ton of money refinishing woodwork.

OK, I'm sold!

Our Home Depot only carried it in an oak finish, so it took awhile before I could locate it in cherry. I finally found some last week and was excited to begin.

From what I understand, this stuff is a special stain that can make its way through the varnish to fix faded/watermarked/otherwise damaged wood. Awesome!

Unfortunately, here's what happened when I tried it:

Polka dots! ugh....

As near as I can tell, whatever my cabinets were varnished with is tough as nails! Each of those little dots is an area where the cabinet was bumped or dented, or had a natural pit in the wood--so areas that the Restor-A-Finish was allowed through the varnish.

waaaahhhhh!

We even tried steel wool, hoping it would pit up the varnish enough to let the Restor-A-Finish through all over.

nope...

Any ideas? We're debating whether using super fine sandpaper would be a good or bad idea...