Similar to a Conan the Barbarian novel, but not that one

I remember reading this hardcover book that wasn't much bigger in dimensions than a paperback, but it was thicker than the average one, maybe because it was hardcover.

There is one part I remember the most was where the protagonist was chasing the antagonist across the desert. It went something like this:

The <antagonist> was fleeing across the desert and <protagonist> chased him. They ran for days, by the third day, he had begun to strip his armor and was turning red under the heat of the sun, but he couldn't give up. I think the armor came off in pieces, and it wasn't knights of the round table armor, but I believe it was more Dungeons and Dragons type buckler or leather armor.

I honestly don't know if that's enough to help, but it's all I can remember. The description of the main character was basically the same as Conan. I had thought maybe it was a Conan novel, but none of my searches have yielded results. I was between 15-20 I guess when I read it. That would be roughly 1986-1991.

I have always wanted to read it again. Thank you in advance for any potential leads and help.