The mortgage and its payoff is irrelevant. What matters is the gain.
The gain on the sale of any capital asset (such as a house) is the amount you get for the sale ($505k here) - the costs of the sale (commissions, taxes, etc.) - your adjusted cost basis. Your adjusted cost basis is what you paid for the property ($149.75k) plus any improvements you made (e.g. adding a room, major renovations, not just repairs). Without any more information your gain is about $355.25k total or $177.625k each.
This TT FAQ will describe in detail the conditions for the "section 121" capital gain exclusion of $250k per person if you have used the home as your primary residence for 2 of the last 5 years. It also talks about how to report the gain in TT. Feel free to ask again if you still have questions.
https://turbotax.intuit.com/tax-tips/home-ownership/tax-aspects-of-home-ownership-selling-a-home/L6t...
I'm a little confused by "we sold the home that was given to my ex to live in." Did you ex buy you out? Or did your ex live in it while it was for sale or while the divorce was pending? If you sold to a third-party it is more straightforward.
Did you get a 1099-S for the sale? If so whose name/SSN was listed?
Finally, the easy case is when you bought the property jointly, each made 50% of the payments, and split the profits 50-50. Anything else can complicate issues. Also relevant is the wording of the divorce decree. Sometimes who pays tax on various items can be spelled out there.
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