I agree with the above comments. I spent some time in Norway during June 2022. Although I am very much a book-it-as-I-go traveler (though usually 48+ hours ahead rather than day of arrival), my pre-trip research backed up what I had read about costs in Norway. I didn't want to be stuck spending a lot for lodging, so I booked most of my hotels at least weeks ahead of time. Oslo was less of an issue than other places, I think because business travel was reduced in the summer, and that's a key part of the lodging demand in Oslo.
You'll have a car, which gives you flexibility to choose lodging across an area, rather than being tied to a specific town. That will help. Another thing that would help is a willingness to stay in apartments or other private accommodations rather than hotels. I don't like worrying about whether the key holder will show up to let me into an accommodation, so I stick to hotels wherever possible.
I'd recommend taking a look at lodging rates on booking.com for the dates you'll be in Norway to get an idea of what you'll need to spend. Not all rooms are listed there, and booking directly is likely to save you money, on average. On the other hand, the costs you see today are likely to increase between now and the dates you'll be in Norway. I took a look at Alesund, just because I've stayed there. For the night of August 28, the lowest hotel rate I found was $178 for a single room. The next cheapest rate was $208.
