We've been to Alaska 3 of the last 4 years and we rarely stay in campgrounds. If you are a member of the YMCA, they have an "Away" program that allows you to use most Y facilities all over the world. You can go to their website and find a Y along your route or where you will be visiting. When you enter the Y, have your membership card/tag with you. Ask if they participate in the program (we found only one that did not) and they will usually ask that you fill out a waiver. They will sometimes call back to your local Y to verify. My husband and I are avid runners/cyclist and we use the Y facilities for our workout and shower.
Another option for showers are community recreation centers. We often use them. An example being the one in Kenai, AK. Depending upon who is working the desk, it was either free or they charged $2.
Some laundries do have showers and charge about $3. We found them to be adequately clean.
Regarding obtaining water, we often got water when we did our fuel stop. For dumping, many towns do have public dump stations. Again, just go to the town website and if it's not listed there, give them a call and ask.