I can't work out how to change my plan. The "manage plan" page only has the option to cancel my plan. There doesn't seem anywhere on the website where I can change my plan, only to buy new ones

Go on the side and click cancel on your current plan after that press "It's to expensive" and you get offered another plan.

It doesn't for me :/ I think it's possible that I've gone through this flow before and accepted a slightly cheaper offer to keep me as a customer. I wonder if that permanently removes such offers from appearing again?

This worked for me a year ago. You can cancel the new one within x days. Couldn't believe it worked, but was happy with it.