Overnighting in eastern Jeju: worth it?

Hi all,

I'm planning a 7-8 day trip to Jeju sometime in July or August, and It's a bit of a struggle (in a good way). Where I'm stuck is deciding where to stay without hopping around too much. I've worked out that I'll want to stay in Jeju city/Aewol for most of the trip, but after that it's a toss up: most people recommended Seogwipo, but a lot of what I want to see more of is in the east.

So the question is, is it worth spending a night or two in eastern Jeju or even on Udo Island? Any advice or suggestions? Thank you!