How are you splitting your time between the parks? Even though it's not as convenient for packing/unpacking, I think it's best to have multiple lodgings to minimize your time driving (and even in Yellowstone people typically recommend staying in several different locations). Also, if you are interested in taking pictures at sunrise or wildlife (better in early morning), you will want to be closer to your destinations.
I also prefer to stay in the parks for the convenience. Amenities are typically basic (which we don't mind - we often camp anyway), but you probably won't be in your room that much. If you want more amenities you will want to look outside the parks - West Yellowstone for Yellowstone. The lodges are pricey inside the Tetons. We stayed once in the Colter Bay tent cabins and never again. We don't mind roughing it, but I was not a fan of the mice. So I would definitely not recommend those! We've never stayed outside the park, so not much help there. I would probably look at Teton Village - it's close to the Moose-Wilson Road which can be good for wildlife spotting. There will be a lot of options in Jackson.
I would highly recommend staying at least one night around Old Faithful (there is both a lodge and an inn). There are a number of geysers here and being able to watch eruptions at sunset, sunrise, and nighttime is really special and easy to do if you are staying just minutes away.
Lodging inside the parks goes quickly so I would book as soon as you can if you think you want to stay inside the park. They have generous cancellation policies, so you can always book now and change as needed.