Now that Shanks is a cripple winning against him would mihawk make look like an idiot. And if he somehow gets defeated or draws with shanks even though he has only one arm the shame is even bigger on him. It's a lose lose situation thats why mihawk doesn't challenge shanks anymore. If he still had his arm he would try to settle things. But now it would only bring him shame if he wins against Shanks.
I believe Mihawk stop fighting Shanks because he doesn't want to beat Shanks by exploiting any weakness, holes, or changes in his swordsmanship because of his missing left arm.
We know Mihawk is waiting for a swordsman stronger than Shanks. That means Mihawk considers Shanks the second strongest swordsman in the world. The databook tells us Shanks didn't lose any power after losing his arm. Based on those facts we know the issue is not because Shanks is lacking power.
In One Piece pure swordsmanship doesn't exist. A swordsman can use any type of power in combat with a sword and fight any way they want. Even though there's no wrong or right way to fight with sword, a swordsman strongest attacks comes from channeling their power through their sword. We have seen that time and time again.
If Mihawk and Shanks spent years fighting each other they probably know each other's fighting style, strengths, and weaknesses better than anybody else. Since Shanks was left-handed and is missing his left arm he had to make some adjustments and changes to his swordsmanship. IE: Shanks had to learn to fight with a sword with his right hand.
I think Mihawk is the only one able to notice the subtle differences between his current swordsmanship and his previous swordsmanship. IE: Shanks Divine departure might be slightly slower now than it was in the past when he performed the technique with his with his left hand. If that is true and Mihawk is able to exploit that to beat Shanks that is not the type of victory Mihawk wants. If he loses because he doesn't exploit that weakness it won't be a true victory for Shanks because he won at the cost of Mihawk holding back. I doubt Shanks would want to win because Mihawk held back.
That's a dilemma Mihawk is in. The best decision was to just stop fighting Shanks and wait a new worthy challenger.
Remember, Oda stated Mihawk's the strongest in name and reality. Mihawk is still stronger than Shanks regardless of the reason. I'm not saying Mihawk is the WSS because he can exploit Shanks' weaknesses. I'm saying that would make beating Shanks easier and that's not the type of victory he wants.