The story strongly implies that Shanks would lose to Mihawk if they both fight as swordsman. However, if Shanks does not fight as a swordsman, he is likely stronger than Mihawk.
Chapter 1058 states that Mihawk has “greater sword skill than even (Shanks).” This statement emphasizes their skills as swordsman, but ignores their power comparison if Shanks does not fight as a swordsman.
It seems that Mihawk would defeat Shanks if they both fight as swordsman. However, by solely emphasizing Mihawk’s superior sword skill, rather than their comparative total strength, it seems likely that Shanks would beat Mihawk if he did not fight as a swordsman, ie including kicking and/or punching.
The chapter does not imply that Mihawk will have weak Haki because strong swordsmanship requires strong Haki, such as when both Oden and Zoro scarred Kaido.
But personally, I’m willing to bet Shanks Haki is superior to Mihawk’s by a decent margin. Especially his Conqueror’s Haki. Shanks’ epithet is literally “Observation Killer” and Mihawk’s observation is likely his strong suit…you know, being “HawkEYE” Mihawk. So Shanks is literally a hard counter to him.
Nobody is saying Shanks is confirmed stronger, but saying Mihawk is confirmed stronger is just as wrong and the evidence is not only on Shanks side but it’s even more so on Shanks side after this chapter and seeing Oda continuing to tip toe around the issue.
But as OP said, Mihawk has greater sword skill. I believe that’s only because Shanks is missing an arm. Story wise it’s said that Shanks did not get weaker after losing an arm (if I’m not mistaken), but after he did lose it, he also lost the prestige he had as a swordsman and maybe the drive to continue striving for the title.
Theory by InreMugiwara (https://www.reddit.com/user/InreMugiwara)