Some posters have given the answer, but others have been distracted by the 180 number that always seems to pop up in these discussions.
Should you increase your cadence: YES
Why?
—Your cadence is relatively low (~165)
—Your legs are significantly shorter than average (5’6” with a long torso)
—Your pace is reasonably quick (6:30, your hour race pace)
—You’ve GOTTEN INJURED in ways that are credibly related to your low cadence
Again, the question is not whether you should be at 180 all the time no matter what. But you have a cadence that is substantially lower than “expected” given the first three factors above. The fact that you have hip injuries is a reason to increase your cadence, but also to do it SLOWLY. The real benefit to a cadence change is the form changes that will naturally follow. So I would recommend that you practice running 6:30 pace at a cadence of 170, and low-160s for your easy runs. (Make sure that this is actually slightly higher than what you would do otherwise). Also, only focus on the cadence increase about 1/3 of the time. This way, your body will “remember” its usual form and not fall into bad or weird habits.
You should also be doing PT for your injuries, and other modalities if necessary. If going up by 5 feels good, you could eventually push it up by 5 more. Try to notice how the cadence difference feels: your stride, your body, your hip in particular. After a couple weeks it should eventually feel more relaxed and controlled; if it feels bad, you may need to drop it altogether and consult a coach.