Hi! How about Lamium, creeping Jenny, Vinca, Mitchella repens, Creeping dogwood, creeping Thyme.

Mazus tends to prefer an area that gets a decent amount of moisture, particularly if it is in sun. The more sun you give it, the more moisture you have to give it. It does not like to bake under hot summer sun. It will choke out weeds once established. Pratia likes part sun but like Mazus needs reliable moisture. I think of the three, Herniaria glabra is the best choice for the beach because your soil is so sharply drained that it won't hold much water and its nutrient needs are not such that it will starve in the sand. You may also wish to consider Sedums, Thyme, Veronica, Snow-in-summer (Cerastium tomentosum), Creeping Crosswort (Phuopsis stylosa), or Delosperma (ice plant) for full sun. Covering dry shade is tough. Perhaps Waldsteinia (barren strawberry) or Lamium would work. Do not take this as the final word as I am not too familiar with beach applications. But, this should at least help get you going.

My go-to ground cover of choice is Big-root geranium (Geranium macrorrhizum). Here's why: full sun-to-shade; moist to dry soil; fills a space within a season but is easily removed just by pulling up; never invasive; pretty bright pink blooms in spring (just finished) with some rebloom; lovely fall color; divides and transplants like a dream; shades out all but the most determined weeds which are usually easy to remove because they generally can't get a firm grip under the geranium shade. I currently am putting it to the ultimate test. I just transplanted some under a large blue spruce, i.e. dry and total shade, next to an open field that receives a direct hit from our north winter winds. I've watered it to get it established. We'll see what happens next!

From another site "Q: I have creeping Jenny in my azala, rose and hydranga beds. It seems to suffocate the roses. Do I need to worry about it suffocating any other plants? A: Creeping Jenny has one of the most extensive root systems on Earth and will smother everything near it. IÂd do something to get it out of your landscape beds if they are of value to you, even if it means hiring a landscape maintenance company to come in and do the work." The vinca and the sweet woodruff both are easily controlled where I live. My vinca grows in a 1 1/2 foot swatch along both sides of my front walk way, I trim a little out every year in the spring, and snip occasional trailer off over the summer. The Sweet Woodruff has such shallow roots that it can easily be pulled by hand.