Do you have some sort of edging around your veggie beds? If not, I highly recommend it before you add any sort of ground
cover.

My other concern is that "various mulches float up and make a mess in the rain". That to me means you have an issue with water run off. It's not running off and draining fast enough which will be hard on any ground cover except for the "dreaded" things you don't want near your veg. garden like monkey grass.

If you can find a way to make the run off more efficient, then you might want to try some creeping thyme between your stones. As long as there is bed edging, many can be successful down here. But it would creep into an unedged bed quite easily. I find that they thyme remains quite thick between stepping stones and I do not have to worry about weeding very much.

GGG

I would suggest to try some dwarf mondo grass. Not to be confused with liriope. This will stay small and grow in tight clumps. It is not overly aggressive and can actually take light foot traffic. I have it planted around a pathway in my garden and it does great. I would agree with girlgroupgirl, and say to maybe putsome kind of edge or even a small metal strip about a foot wide down between the garden and path just to keep runners out of the garden.

Here is a link that might be useful: mondo pic

How about Mazus. It grows about an inch tall, can take wet areas,is evergreen and has really nice flowers.

try a creeping thyme or some culinary herb ...when you walk on it it will smell nice as well...and use it when you cook!

How about dutch white clover?

I found mazus at landscape supply last summer. I believe it was also at Pike's, but it was a little pricy there.

Thanks for the suggestions. I have a rock edge between the paths and the beds on the outside, and then there are raised beds in the interior of the garden. However, the rock edge is really just a line of 6-10 inch rocks, not stacked, nothing buried.

As for the mulches floating up, I have worked the past three years at amending the soil so it's not the solid rock clay it once was, which has improved the drainage vastly. At this point, I just don't want to deal with mulch and having to replace it as it breaks down. In addition, one of the reasons the water pooled up there during the hard storms is because there's nothing growing there to use the water.

We've got the dwarf mondo grass on the other side of the house, so I'm not sure I want to use it again. I've been considering thyme, but am concerned about it's aggressiveness. I've never heard of mazus, but will look into it.

Thanks for the suggestions.

Mazus is really nice. Another thing to consider is blue star creeper (also comes in white star creeper). I use it mostly in a bed that has drainage problems, but it has worked for me wherever I plant it. It's spectacular when it's covered with the tiny blue flowers (although in my experience the blooms last longer with some shade). Golden creeping jenny is also a good cover. In some places I've planted it with the blue star creeper and let them fight it out for coverage. It makes a pretty combination.