St. Augustine pallet in BR - good idea to lay down soon?

You can lay sod almost year round. But definitely no issue putting it down now or within next couple of weeks.

$600/pallet is ridiculous. St. Aug pallet is about $250 or so. If you don’t have a truck/trailer, I’d expect delivery to run no more than another $100 within a reasonable delivery.

Woerner Turf in BR usually has good quality sod and I know will deliver it or you can pick it up.

You may read that sod should not be laid in Oct/Nov or later, but that's wrong for the Gulf South where cool months are ideal. I had St Aug sod put in Jan/Feb a few years back. The pallet went through an overnight hard freeze and the freshly installed sod had 2 frosts. In spring the yard greened up nicely and is quite healthy now. Sod bought in winter will be about 50% brown leaves, which is OK. Buy from Woerner not a garden center that caters to people who want just a few feet.

Other than watering I'd recommend getting a lawn roller - LINK

I rolled the new sod right before frosts to make certain there was a soild contact with the soil. I also roll any areas where edges were beginning the curl.

Sod fails most of the time because it drys out or it's put down on hard compacted bare ground... till if needed.