Mihawk fights in stages.
If he thinks you're too weak, he only uses the knife (EB Zoro).
If he thinks you have skill, he uses Yoru (MF Vista).
If he thinks you're strong enough, he uses haki through Yoru.
And if he thinks you're truly on his level, he goes all out.
I'm proposing that Mihawk only went as far as lvl3 against Shanks, which Shanks could take it. The next step would've been to go all out, but they were interrupted, and by the time they met again, Shanks was lacking an arm.
If being a one-armed swordsman did not make Shanks any weaker, as the manga has told us, then why is Mihawk refusing to duel again?
Was it perhaps something to do with what that missing arm gave Shanks access to?