Since you're moving to Utah permanently, your domicile will be Utah after the move and therefore you will be required to pay Utah income taxes on income you receive from the day you become a Utah resident. That would include the capital gain from the sale of your TX principal residence.
I'm not in Utah, but I would think Utah conforms to the federal Section 121 rule on a sale of principal residence where a seller can exclude up to $250,000 in capital gain if s/h had resided in that house for 2 out of the previous 5 years (married couples get up to $500,000). I did a quick search and found this article. But I can't say if it's accurate. The article implies Utah conforms to IRC 121, but doesn't say it explicitly. So you should check it yourself. If Utah conforms to the Internal Revenue Code and your gain is less than the Section 121 exclusion amount, then I don't see any reasons for delaying your move.
https://www.deseret.com/utah/2022/4/8/23013440/housing-market-tax-selling-your-home-capital-gains-ta...
Here's a copy of Utah income tax return, TC-40, it starts with federal AGI, so it implies Utah does conform to the IRC, but you should check that yourself.
https://tax.utah.gov/forms/current/tc-40.pdf
Utah part year tax return, TC-40B. This is the form you will file for 2023.
https://incometax.utah.gov/instructions/tc-40b
