Here is a picture with the grass burned out.

Knowing "of" a plant is not the same as having grown and lived with it for a period of time so I can't speak to how well creeping thyme will likely do for you. However, that something is "drought resistant" does not mean that it will be as easy to maintain as concrete. Almost all plants require a babying period when they are newly installed and their drought resistance capabilities are limited, too.

Do you have irrigation at this area? That is how most people solve the grass-browning-out dilemma. If the grass here is contiguous with the rest of your lawn, keeping it, as opposed to changing to another altogether different looking plant, may look the best.

We can't see what's to the right and left of the picture you've posted, but in general, the house looks like it might have a very exposed look. Ever consider putting a tree at each side, some distance away from the house, in order that the setting might look a bit more sheltering of the house?

I don't have an irrigation system. I am trying to save on water usage, it's getting expensive. There isn't enough room on either side of the house to put trees. I had a few on left side but they got too big even though they weren't supposed to grow that big. I had them cut down years ago. There is a dogwood just to the right of the area I want to do over and there are plantings under that. The tree on the left was cut down many years ago too when it started to die. I am sort of exposed on the corner also. I am thinking of putting in small shrubs and perennials and then mulch that area. The entire lawn took a big hit last summer when we had a pretty good drought going on.

I think changing the grass area to small shrubs and perennials is a great solution! Keep in mind the need for evergreens and structure, and have some fun with creative design.

One other thought on this is that, if one is having to reseed on account of grass drying out every year, one is also NEVER working with ESTABLISHED turf, meaning that it never has a good root system ... and therefore is even more likely to suffer at even minor droughts. Sometimes plants, including turf, must be babied in order to get them to be well established. The "babying" time might be for the first two or three years. If turf requires too much coddling then you may need to move on to plants that require less.

When you say move on to shrubs for this area, the devil will be in the details. Personally, I loathe shrubs used in a groundcover kind of way because it adds too much to the maintenance, keeping a mass of shrubs top trimmed. A groundcover or perennial used as a groundcover seems to offer greater possibilities. Whatever you do, though, it'll require some supplemental water until the plants become well established.

I don't mean groundcover shrubs if that's what you are saying. Maybe just a something dwarf towards the front.

YOu would have to spell out an example because it sounds like dwarf shrubs used in a groundcover-like way. I'm not sure what "groundcover shrubs" are to you.

I thought you meant something like the creeping yews.

I had nothing in mind but was responding to your suggestion of using "... dwarf shrubs toward the front ... " I don't know how you can use them to replace grass that is any other than like a groundcover because, presumably, there would be several shrubs next together. I'm only warning you of a potential danger of doing this. Maybe you love the idea of trimming more trimming tops of shrubs. I don't know. But I know I wouldn't want to do it. I'd rather just trim the edge of something that was creeping over the walk.

You wouldn't believe how many shrubs my other half trims in this tiny yard. He is very vigilant.

Consider short ornamental grassy looking species of groundcover, such as liriope or maybe some of the sedges, if any grow in your area.

And drought tolerant or not, you HAVE to water plants regularly until they get established, which is usually the first year or two.

"You wouldn't believe how many shrubs my other half trims..." Then let's give the poor guy a break!

I would use a sedum rather than any of the thymes. Sedum ' John Creech' would be perfect.