I haven't used Pandora in a while, but I think the stations you use should be properly linked to your account.
So first troubleshooting steps: triple check that you're using the exact same account (I know you said you are, but checking again never hurts). You could also try to add details to the stations you have, and that could make your stations get 'remembered'.
And on the PC that DOES work, try to log in using a different browser (or using private mode/incognito mode), to see if it's just a weird thing that happens only with the one computer.
Then I would say, if that doesn't work, share all the stations you care about (Pandora has a built in sharing function that hopefully will keep it exactly as you want it), then log out and log back in. Maybe that will force Pandora to save the stations. If Pandora happens to forget, then you might be able to restore the stations because you shared them.
You can also try to ensure you have Pandora logged in on only one device at a time, in case Pandora is weird about that.