We had a florescent lighting nightmare in my 1989 kitchen as well - we ended up doing can lights in the soffit, and in the middle we hung two big lanterns over the island. I actually don't notice shadows at all with the can lights, they brighten things up more than I thought they would.
Before:

We went with a pair of Bessa 3-bulb ceiling fixtures for center of room. They have a collar which I intend to paint with the ceiling paint. They screw on and off the collar so I won't have to futz with the wire attachers that many such fixtures have when I need to clean out bugs or change bulb.

Matching smaller ones in adjacent hall and lobby. We chose a wonderful matte white carerra? glass in them and they coordinate well. More lighting -- pendants and undercab -- over the work surfaces.

Not common on the GW but it works for us.

Here is a link that might be useful: Besa ceiling lights

My cabnets don't have a very deep lip either but I put the under cab lights there anyway. I've always had lights there in every kitchen I've ever had so it was a given I'd have them here. Mine are pretty thin and don't really show either when they are on or off. You might be able to put a thin enough molding on the bottom of your cabs to hide most of the light fixture. I wouldn't be without under cabs. Will you have lights over the sink?

Another idea that might work or you might be able to adapt it. Where your ceiling lights are now, finish that space and put some good lights there. If you fancy it up a bit it could look spectacular. Maybe rope lights? Or rope lights under the cabs?

Here is a link that might be useful: a ceiling like this?

Thanks for the suggestions, especially the pictures. I think the can lights might work after all.
I know everyone likes pictures so here is my kitchen. Yes, there's a lot more that needs to be done besides the lighting but my budget is small and my patience for redecoration is even smaller. I'm in awe of the creativity and attention to detail I've read here!

I like tile, wood, and stone; I like rustic earthy colors; I'm fond of the Arts & Crafts look and I also like Spanish/Mexican tile.

With can lighting, the trick is to place them in front of the person working in order to prevent shadows. My contractor suggested placing them around 20" from the wall, just in front of a person working at the 24" counter. We also have a few in the center of the room too.

We have CREE LED can lights (2700 K) and the light is very warm, not harsh like fluorescent lights.

I replaced my recessed fluorescents with Cree LR6 as I had the ceiling raised by ~ 1 ft.

There are lots of very thin fluourescent undercabinet lights available that are barely visible that you could use.

I also have seen some light fixtures that might work for you at brodwax.com. and fluorescentgallery.com. Some of the other websites that I looked at for Title 24 lighting are croftandlittle.com, ylighting.com, junolightingroup.com, lumens.com and ceilingfixturelighting.com.

Once you start looking around, I am confident that you might a few possibilities.

Good Luck!