It is not really normal. Bettas are more of a hovering fish, if he stayed in one place all day, I would be less concerned, Its great that he is not nipping his fins, but he definitly sounds stressed.
Is the temperature stable at a temperature between 78-82 day and night?
betas are aggressive to other betas, and fish that they mistake for other betas . but that doesnt mean that they want to live a life of solitude. Many beta fish often thrive in tanks with carefully selected tankmates. You might consider adding somthing, maybe 6 harlequin rasbora, or galaxy rasbora, or ember tetra. unfortunatly options are further limited with a ten gallon.
If you decide to do add more fish, do it on a day and time when you can watch them often. keep a back up plan in mind. worste case senario is , you have to give the new fish away.