I am not in either of your areas, however I have seen this situation accomplished many times.
This can be difficult to pull off, but the easiest way to do it is to make your purchase contract (on the home in San Antonio) contingent upon the sale of your home in Utah. This means you won't and are not obligated to settle on the new house until your previous one sells.
The challenge is that some home sellers will not accept 'contingent' contracts... and want you to buy it regardless of whether or not you sell your other home. Have a good local San Antonio real estate agent help you pick houses that would accept this contingency.
If your income is high enough you can sometimes purchase the new home and still keep the old one until it sells (obviously you would have to make both payments)
You can also possibly obtain a 'bridge loan' - these can be dangerous types of loans to get into. They are not common much anymore, but were popular back during the 'boom' years. These loans are set up so that you can purchase a new house while avoiding a payment for a certain amount of time... and hopefully within that time you will sell the house. If that's the route you really want to go, talk to some local lenders and see if they offer it and get all of the finite details as they can change from lender to lender (many don't offer them any longer).
Your best bet is the contingent route - but you want to make sure that your Utah home is priced to sell because even if a seller accepts your offer, they will usually have a 'kick-out' clause attached to it... That basically means that your contract to purchase could be kicked out if they get another offer they like better. They will give you (x) many days to remove your contingency or they will cancel your contract and go with a new one.
It can be difficult, but certainly not impossible. Get good local representation in each area and coordinate it with them - you should be fine!