Before writing this reply, I tested this again, to make sure it still works. I switched my Steam to offline mode, closed it, put my computer in airplane mode to disconnect it from the net, started Steam, and then started Conan Exiles. I used BUGLE instead of the official launcher, but that really should not make a difference.
The failed to login message I got was this one:

Clicking on “OK” left me in the main menu, and then I pressed “Singleplayer/Co-Op”, followed by “Continue”, and I was able to play my single-player game.
Is it the same message that I posted above or something different? And what do you mean by “stuck”?
What exactly went wrong? Did it give you some error message when trying to continue the game? If so, what did the message say?