Yes, you should fertilize in the fall - but I think it may be too late to do it in your area. September is the month in central Texas to apply a good 8-2-4 organic slow release fertilizer. However, any time is good to put down a quarter inch of good compost, so you could do that now and it would be helpful to your lawn.

Yes, fall fertilization is probably the most important application for a decent lawn the following year. There is no need to purchase expensive mixes and definitely steer clear of anything with phosphate in it (the middle number) unless you have done a soil test and have a known deficiency. Potassium is nice but not necessary. It is generally considerd to improve stress in regards to tolerating summer heat.

A simple bag of ammonia sulphate (21-0-0) from the feed store will perfom admirably. Put down between 4-5lbs of actual fertilizer per 1000sft of turf. This will give you between 0.8 to a tad over 1lb of actual Nitrogen per 1000sft of truf.

Dittos to the fall fertilization. But you should get right on it.