There are free showers airside, but the ONLY way to access them is by transiting. You won't be able to get past the transit security area without a boarding pass. In other words, you can't play dumb and get to them.

If a shower is that important, seriously, your only option inside the airport is the Spa on Air B1, if you're arriving and staying in Korea. The Grand Hyatt "arrivals lounge" is also kind of a misnomer. You have to go the hotel by shuttle, get there, and use their spa facilities. And it is/was only for KE Business/First passengers. In fact, you could just get going into Seoul.

For example, land at 5, get out of the airport by 6, get to the sauna by 6:30, Shower and freshen up (30 mintes), go back to the airport 7:15, bus or subway 8:30 arrival to Seoul. That's a conservative estimate.