Do you have AWOL? I just finished my thru hike and I don't feel like it was too long between showers for me.
Boiling Sings PA- Shower at Pool
Duncannon PA - You can pay for just a shower I think but rooms are only like 20 or 25 bucks, laundromat in town
Port Clinton PA - You can pay for a shower at the hotel/bar, they might have laundry there? check AWOL
DWG - Donation based Shower and laundry at the church hostel
Vernon NJ - Donation based Shower and laundry at church hostel
Dalton MA - Shower and laundry at a trail angels house, not sure if he'll be accepting hikers in this year though but AWOL will tell you, there may be shower and laundry at the nearby church hostel as well, I just didn't stay there
Manchester Center VT - You can probably pay for shower and laundry at the hostel
Glencliff NH - Don't miss it, shower, laundry, and a chilled out palce to relax before the whites.
North Woodstock - Trail Angels house,shower and laundry. ask around
Rangeley ME - I think you can pay for just a shower and laundry but it's worth it to stay here
Caratunk ME - SHower and laundry at Sterling Inn
Monson Me - Shaws or the Lakehouse
There's tons of other towns and places to do these things, this is just what I did. All of the top of my memory, crazy how it sticks with you
My best suggestion is to buy Awol and formulate a plan for yourself, or just figure it out as you go. The trail will provide.