PNGs look washed out after saving them from Photoshop

IF you have a file that’s "sRGB IEC61966-2.1 and Photoshop converts it into Adobe RGB (1998)" then you can fix that behaviour in 'Color settings': 'Color management Policies': select "Preserve Embedded Profiles" there.

Viewing images in applications that do not have colour management built in cannot be expected to match Photoshop's fully colour managed appearance. 

You can assess screen accuracy visually to some extent using a reference test image: please go here and download the Adobe RGB testimage: https://www.colourmanagement.net/index.php/downloads_listing/

I use this to assess accuracy: Have you ever wondered how to KNOW whether your screen [or printer] is ACCURATE and not just 'pleasing'?
If so please check this out: http://www.colourmanagement.net/products/icc-profile-verification-kit

I hope this helps neil barstow, 
colourmanagement net :: adobe forum volunteer
google me "neil barstow colourmanagement" 
for lots of free articles on colour management

...and make sure that the profile is embedded!

With an embedded color profile, in a color managed application, any profile will be correctly represented and the files display identically.

Windows "Photos" is now fully color managed and will display identically to Photoshop as long as there is an embedded profile. Without a profile, all bets are off and it can look like anything, depending on other factors.

In other words, it doesn't matter which profile. It matters that there is a profile, and that the application reads and uses it. The choice of which color profile to use is determined by other considerations. That said, the safest choice is usually sRGB.

After you save both, you view them where? IF that other application isn't color-managed, that is the problem.

Without color management, sRGB nor Adobe RGB (1998) has any meaning.