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.

None of those GC's is particularly efficient with regards to weed control and blue star creeper can often die out in my zone 8 winters (at any rate, has very erratic growth patterns).

I'd look to beach strawberry, Fragaria chiloensis, or creeping rubus or creeping bramble (Rubus calycinoides, syn. R. hayata-koidzumii). Both are quite happy in sandy soils, will transition from sun to shade and grow quite rapidly. The rubus is one of the best weed control groundcovers I know of, as well. Waldsteinia is a good suggestion too.

I have been doing a little reading about the creeping rubus/bramble/raspberry and it is starting to sound like it would be my new best friend...I have read nothing but POSITIVE COMMENTS about it.
However I haven't read anything about thorns or prickly vines...an errant grandchild may wander barefoot into that area on occasion. Does this plant have thorns?
Also, any suggestions as to where I might find this plant at a price more reasonable than $5 for a 3" pot (not including shipping). It seems almost sinful to pay that much...wish I had a neighbor with this plant who wouldn't mind giving me a spadeful. Perhaps I can find someplace online that ships it "barefoot"

That would be "ships it bare-root", not bare-foot haha

Creeping Bramble (R. calycinoides/pentalobus) doesn't have thorns, but it is somewhat painful/rough to touch -- the stems are somewhat coarse textured. I usually wear gloves when I handle it. Not sure about walking on it.

If you go with it, the 'Emerald Carpet' selection has brilliant red leaves in fall.

Also, disagree about Rupturewort (Herniaria glabra) not being a good weed suppressor -- it forms a very dense mat so it hard for much of anything to get through.