I remember when Prince Andrew sprayed photographers with paint--I think that it was in Los Angeles--in 1984. That sort of killed his post-1982 honeymoon with the public. Then his stocks went up again when he married Sarah, because the two of them seemed to be obviously in love. In fact, they literally clung to each other, being VERY affectionate during the Canadian tour in 1987. For quite a while, Sarah got the blame for their decisions such as leaving Beatrice behind in the UK during the 1988 tour of Australia and having their family photos in Hello! Andrew was pretty much out of sight during this time because he was in the Navy, which Sarah blamed for the failure of her marriage.
I think that it's a combination of factors that keeps them together. Perhaps Andrew feels responsible for Sarah because he wasn't around while she was trying to adapt to Palace life, and then because his salary wasn't very high when they divorced. Plus there's the fact that she's never publicly "dumped" him. It's almost like the separation and divorce was simply a legal thing and didn't happen emotionally.
I would be shocked if Andrew took up with another lady long-term and married her. He said in an interview that his regret was that he wasn't able to hold his marriage together like his parents did. So having failed once at marriage, I don't think he'd take another chance at it.