You didn't state your zone, so could be that they're drying out too much or too hot of roots in the pots.

Try growing them directly in the ground or transplanting them to the ground sooner. They need nutrients & room for roots to spread. If they dry out in the pots they're toast.

I grew herbs in pots for years, but with limited results from seeds. Used a variety of pots small & large, plastic, clay, glazed, metal tea pots, etc. Once I moved those same herbs into the ground WOW they spread a lot & the flavor improved. Probably because the plants were healthier. I've kept some in pots just for looks, but they're not thriving. The ones in the largest pots do okay. Oregano is growing well in a metal colander & a large steamer like you'd use for clams or pasta. Kind of funky, but I like the electic look.

Also, have parsley in a plastic pot inside a large enamel pot with a tea cup & saucer on top as well as a large kitchen spoon & server holding the lid off to the side.

Which herbs are you trying to grow?

FataMorgana

sounds like you are watering too often to me... try watering more heavily less often.

most herbs should do fine in pots...

Corrine1, I live in Coastal Southern California. I tried to grow the seeds directly in the ground, but I never saw them to some out. I think something ate them right after they came out. We have squirrels, rabbits and also snails in our yard. Rodents too.

FataMorgana, I try to grow different varieties of Basil, Sage and Lemon Balm.

I have tried to grow perennial rosemary bushes two seasons in a row without success. This year's bush was planted after Mother's day in fertilized garden soil in a garden that gets 5-6 hours of sun each day. It was kept well watered and thrived through July. Now at the end of July it turned pale and died over a two week period. Rose bushes nearby are doing well. Any suggestions for next Spring?

Corrine1 - I had some chives growing in a clam steamer for a while, but they died, I think a dog got in the fence & peed on them.
A neighbor gave me a bunch of containers from his barn, in with them was two OLD nail barrels. I cut holes in the side of one to make a sort of 'strawberry pot' for my spearmint. The other has a spiderplant in it. Then dh almost flipped when he found out I was offered $70 a piece for them lol I like them better as planters ;)

Pagirlmint- Sounds like you've tried a lot of things, but here are some more ideas for herb success.

Make a protective covering for the inground planted seeds so that the 'critters' don't get to them.

Improve the soil with some compost to help retain some moisture.

Create a raised bed closer to the house & possibly with a fence or chicken wire structure/cage to protect your herbs. I've even seen wire or plastic baskets staked down as protective cages.

You said you were trying to grow basil, sage & lemon balm. They like different growing conditions.
basil: likes your summer heat with rich, moist soil
sage: good drainage, lots of sun
lemon balm: part sun, rich, moist soil

I haven't had much luck growing those in pots, but they all do better in the ground in different locations in the garden. Once you get sage & lemon balm growing well both should overwinter for you to greet you with fresh spring growth next year.

Hope that helps ~ Corrine