The websites may no longer remember you if Spybot removed cookies. Spybot scans for usage tracks, including cookies, and some of those cookies probably contained your passwords, etc. There's a Usage Tracks page here. It says at the bottom it's an older page, but it should still be useful to tell about Usage Tracks:
https://www.safer-networking.org/faq/usage-tracks/
So, you could stop scanning for Usage Tracks altogether, and that should help keep you logged into the sites that ordinarily remember you in Firefox. To do that:
Rightclick Spybot Start Center and select Run As Administrator. If it opens in Overview mode, click Show Details in the lower right corner. Place a checkmark in Advanced User Mode, click Settings, go to the categories tab, somewhere in the window rightclick and select Spyware scan only.
Or if you prefer to keep scanning for usage tracks,you could try getting Spybot to ignore Cookies instead. Rightclick Spybot Start Center and select Run As Administrator. If it opens in Overview mode, click Show Details in the lower right corner. Place a checkmark in Advanced User Mode, click Settings, click the Ignore List tab, then click the Products tab listed there and click the Add...button, and in the space at the top, type in Cookie, then select Cookie below, press Ok, and after you see Cookie added to the Ignore List, press Apply and Ok.
I hope either the first or second option helps.