Finding public shower in Europe

EDIT: Just noticed you said you don't intend to go to a swimming pool but that is my first suggestion. You don't have to swim or anything to use the facilities, only pay the entrance fee. This is probably the cheapest and most common shower facility but I have other suggestions too:

For Europe, I would go towards water. Pools are quite common to find and the entrance fee is cheap. I have never seen one which did not have showers available. Spas in Hungary have showers and some charge very little for a stay under 3 hours (you often pay for the day and get money back when you get out under a certain amount of time). Iceland has thermal baths in nearly every town and have shower. Except for the large tourist ones such as the Blue Lagoon, entrance is not expensive.

Next place would be along beaches. Some offer only outdoor showers which is obviously not ideal to fully wash but ones in built-up areas sometimes have small service locals with private showers and lockers.

One idea that comes to mind is to go to a gym. I have never tried one in Europe but in many places they can sell you a day pass for the equivalent of a few dollars. They may even have a free pass for first-time visitors.

If you have a (rental) car, a good option would be to go to a large fuel station along the highway. That is where truckers shower. I've never used one, but according to a Dutch truckers forum it costed a few euros back in 2011 (they actually complained about euro 3,50 being absurd). This price was mentioned for Germany, France and Italy, so will probably be the same in the Eurozone. Prices will have gone up in the last 6 years, but inflation in the Eurozone has been very low, so expect to pay less than 5 euros now.

EDIT: Hitchwiki confirms this for Germany:

There are showers in most Raststätte and Autohof service stations on and near the autobahn. Prices are between 2 euro and 5 euro.

In Paris and a few other cities, you have public bains-douches (baths and showers), intended for people living in apartments/rooms without a bathroom and homeless people. You tend to find them in small streets in low(er)-income neighbourhoods, you can find a map for Paris (where they're now free) here.

I'd suggest Total Service Stations. The ones on the main highways have shower options for long haul truckies. You pay 2 euro for a hot shower. I'd recommend plotting one on your GPS then checking it has shower facilities on its correlating website (as the ones closer to town are either self service or regular stations). We've done it in Belgium, France and Italy.