Artificial turf can be laid in early spring or late winter i.e. when the ground is not dried and growing period is approaching. Do not lay turf in early winter. The best turf to lay in winter is the village green. The maintenance is less and the durability is high. Even in frosted climate, village green sends root deep underground and it repairs itself. So you may lay the sod in late winter or early spring.

Here is a link that might be useful: Choosing Right Turf

St Augustine can go down any day of the year. When it is dormant you have to be careful not to move it around until it comes out of dormancy and the roots start to knit into the underlying soil.

If you now have a lot of sun, get Floratam St Augustine. It loves the sunlight. You only need to water it once a week in the summer heat. Mulch mow at the mower's highest setting and fertilize on Memorial Day, Labor Day, and Thanksgiving.

Put the sod down now, water it 3x per day for 2 weeks until the roots grow into the underlying soil, and mow when the grass is 5 inches high. Mow it back to 4 inches (mower's highest setting).