Consider a solid (ie: grey) to add contrast and will provide more choices and price point as well.

Tie in the grey by adding cushions on your couch.

If you want more contrast and interest, add art and a rug.

how about something like these?

a blue or turquoise is the complementary color to the saltillo/and leather tones

https://xanadusantafe.com/product/chopper-recliner-southwestern-recliners/

Lone Star Western Decor:

saltillo, leather and black look amazing together!

Wayfair:

and you can get these arm chair protectors in the southwestern pattern. they will fit over most chairs.

this is from walmart

bed bath and beyond

I like Maureen's suggestion. You can then add some Southwest fabric via a throw pillow, throw blanket, or headrest cover. You can cut apart your current chair before throwing it away and salvage the fabric for a pillow cover.

Or go looking for a Southwest upholstered chair like this.

Can we see the sofa? IMO those giant chairs are awful so I need to see the sofa style since without it we guess. I think adding stuff on top of that fabric make it look worse. Just a pic of the sofa and better yet the room.Any comfy recliner can be upholstered in South West fabric first you find a recliner that is comfy .

This is the sofa.

We really want recliners. Currently have the chairs and ottoman situation. Also don’t want to be sitting on leather. Too hot here to be comfortable. Comfortable is the goal.
I was reluctant to photo the sofa and room, and get the ‘ugly’ remarks that already have shown up. Ours is desert southwest style in the desert southwest.

teal fabric recliners will look lovely with sofa

The sofa pic is great thank you. I have to say my comment was not nice I guess but those cahir and 1/2 were just weird always. But the rest of my comment stands you go find chairs you love then the fabric. The space looks a bit tight as is so any recliner smaller then those chairs will be better

If you go to a higher end furniture store, some of the brands they carry will allow you to pick from a wide selection of fabrics, some of which will have patterns/colors.

Many of the higher end brands will also allow you to do a COM order with their upholstered furniture. That means Customer's Own Material and you can pick any fabric, but you would need to make sure it is the appropriate weight/durability for the furniture and intended use.

I ordered/bought 2 sofas from Hickory Chair recently and went with a COM fabric. It does cost more. None of the many fabric options they offered were the right color or durability, so I found a performance fabric in the right color and the design center sent it to the Hickory Chair factory to complete the order.

Here is an example on the Hancock and Moore website of customization you can make to a recliner.

https://www.hancockandmoore.com/YourWay/Motion2

Here are the H & M standard fabric selections ... but you need to make sure it is durable enough for a recliner. Some fabrics are pretty for a rarely used accent chair, but useless for a recliner used every day.

https://www.hancockandmoore.com/Products/Fabrics

You can also use any of the standard fabrics from the sister brands under the Rock House Designer Brands parent company.

https://www.hancockandmoore.com/Documents/Documents/RHF_Fabric_Transfer.pdf

The room looks like it is all beautifully coordinated, so I can see why you want to choose the recliners carefully. I suggest you start by looking at the recliners at La Z Boy since they have a good selection of styles and fabrics. Consider getting electric recliners since you know how awkward it can be to get out of a recliner when you are recovering from surgery.

If the pillows detach from the sofa, take one with you to the store to coordinate colors. You can also call to ask if the local store will send fabric samples home with you for a day or two. La Z Boy will sell you an extra yard or two of fabric so you have spares in case the seat or back fabric needs replacement (cats, dogs, etc.).

https://www.la-z-boy.com/b/living-room/recliners?intpromo=header.Recliners

What about a Morris-style recliner with a SW fabric? https://www.dutchcrafters.com/Amish-Trinity-Mission-Morris-Reclining-Chair/p/46673 

Or Stickley for a more expensive option:

https://www.stickley.com/collections/mission-morris-chairs-collection/products/spindle-morris-chair-with-loose-cushion

Thanks to all for your contributions. I have started looking through your options.

I was unaware you could substitute your own fabric, which is interesting. Unfortunately, I’m 2-3 hours from a major city, Las Vegas, or about 4 to Phoenix, so my options are limited.

The room is approximately 19’ x 18’. The large chairs worked in the space until the recliner moved in.
We do have a high end furniture store in town, which is where I started. They weren’t very helpful, but I’ll try again armed with more questions. They do sell LaZBoy.

Thanks again.