Pretty easy to tell if you are using the Epson driver. Select print from the PS menu, then click Print Settings ...
From there click the popup labeled Layout and you should see Printer Settings. Once you select that you should see all of the choices for your printer, especially all the Epson Paper types. The AirPrint driver looks quite a bit different.
Letting Photoshop manage colors is not difficult, the ICC profiles are all loaded into your computer when you install the driver. Just select Photoshop Manges Colors, click the profile menu and scroll to the one for your paper. If it's not there another clue the driver hasn't been installed correctly.
It's not uncommon for someone not implementing a full color managed workflow to have things break when they switch computers. Something about your old system didn't move across.
the 3800 driver is fully compatible with Mac OS High Sierra.
Not sure how you migrated, but you might need to clean things up, Andrew Rodney describes it as an "Epson enema"
-Download and run the
Epson Uninstaller Utility from Epson
-Open Library>Printers and toss the Epson folder thats in there (this deletes anything epson related)
-Open the Mac Printers and Scanners preference panel (on older versions of Mac OS it's called Printers and Fax I believe) , and control click in the box with your list of printers. Select reset printing system
-Restart the Mac
-Optional: Open disk utility and run Disk First Aid
-Download the current Epson Driver from Epson and re-install the Driver.