Cleyera will take full sun just fine and it is full to the ground very dense, full and a wonderful hedge shrub.

And BTW, ligustrum is not invasive... Privet, however, is. Sometimes ligustrum is called glossy privet. Wax leafed ligustrum or curly leafed lig. would also make a fine hedge.
Go to a nursery in your area that specializes in woody ornamentals. They will be able to show you the plants they recommend...
Good luck~

My mistake. It is the Chinese privet, Ligustrum sinense, and the common or European privet, Ligustrum vulgare, that are invasive in Mississippi. The link below gives the page of invasives beginning with the letter 'L.' As you can see many ligustrums are invasive in a lot of the southeastern states, which is why I posted. In my opinion, if it's not native and invasive in a state near mine, then chances are it will become invasive in mine.
If you'd like to check on a plant before buying it, the home page to find the entire list is:
http://www.nps.gov/plants/alien/factmain.htm#pllists

Here is a link that might be useful: invasive non-natives beginning with L

Waxleaf ligustrum (Ligustrum japonicum) IS invasive in many southern states, including South Carolina.

Privet is just a nickname for ligustrum. If you look at the link carolinakate provided, you will see that it is one of the invasive ones.

What is the botanical name for curly leaved ligustrum?

Cleyera is a good idea - neat looking, and ok for that soil. Sun or part shade is ok if you keep the roots cool (grass or mulch).

Trust me, ALL Ligustrum is invasive, I've killed plenty in the wild.

curly leaved ligustrum= Ligustrum Recurvifolia

In Mississippi, Cleyera does better in part shade. It needs a lot of water, especially the first two years.

Apparently the curly leaved variety is just a cultivar of the japanese one: Ligustrum japonicum ÂRecurvifoliumÂ. Should be considered as invasive as regular japanese privet (usually called wax-leaf ligustrum).