I've got a lot of shade in my back yard. I've sodded with Raleigh with limited success. The amount of sun it gets, approx. 4 hrs, is at the very minimum required of any turf grass I've found.

Is palmetto noticeably better in the shade? Are there any pitfalls to palmetto?

Here is a good article about Palmetto St. Augustine.
https://aggie-horticulture.tamu.edu/plantanswers/turf/staugust2.html

You are going to have issues growing any turf grass in a shaded area but of all the choices, Palmetto seems to be one of the best options. I would consider an alternative to turf grass in a shaded area though as it will never look as full and lush as you are probably hoping for.

Yeah Raleigh has done so-so in the area in question. I'm wondering whether I'll notice any improvement if I went with palmetto.

If not, then I'll just stick with what I have.

Raleigh is an abomination of a St Augustine variety. Give the Palmetto a try

Amerishade
https://krturfgrass.com/product/amerishade-st-augustine/

And if the deer eat it, it's landscaping.

Anyone know of a good source to buy a couple pallets in dfw?

I've had Palmetto for years. My take is that it doesn't really hold up to the shade in any measurably better way. But it does seem to be hardier drought wise, which makes it worth it. It may hold up better not so much from shade tolerance, but just in ability to remain healthier in general

King ranch turf grass has a location in Arlington that I got a pallet of st Augustine from last fall. Will get another pallet this spring.

King Ranch Turfgrass has locations in Arlington, Benbrook, and Prosper. Or Amazing Creations in Colleyville.