You can check the connection settings and check for possible issues with a Proxy or VPN service.

• Settings -> General -> Network: Connection -> Settings

If you do not need to use a Proxy to connect to internet, select "No Proxy" if "Use the system proxy settings" or one of the others do not work properly.

See also:

• https://support.mozilla.org/en-US/kb/connection-settings-firefox
• https://support.mozilla.org/en-US/kb/websites-dont-load-troubleshoot-and-fix-errors

You can try these steps in case of issues with webpages:

You can reload webpage(s) and bypass the cache to refresh possibly outdated or corrupted files.

• hold down the Shift key and left-click the Reload button
• press "Ctrl+F5" or press "Ctrl+Shift+R" (Windows,Linux)
• press "Command+Shift+R" (Mac)

Clear the Cache and remove the Cookies for websites that cause problems via the "3-bar" Firefox menu button (Settings), click the padlock icon (Clear cookies and site data) for the domain in the selected tab.

"Remove the Cookies" for websites that cause problems:

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Data"

"Clear the Cache":

• Settings -> Privacy & Security
  Cookies and Site Data -> Clear Data -> [X] Cached Web Content -> Clear

• https://support.mozilla.org/en-US/kb/clear-cookies-and-site-data-firefox
• https://support.mozilla.org/en-US/kb/how-clear-firefox-cache

Start Firefox in Troubleshoot Mode to check if one of the extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) or if hardware acceleration or if userChrome.css/userContent.css is causing the problem.

• switch to the Default System theme: "3-bar" menu button or Tools -> Add-ons -> Themes
• do NOT click "Refresh Firefox" in the Troubleshoot Mode start window

• https://support.mozilla.org/en-US/kb/diagnose-firefox-issues-using-troubleshoot-mode
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems

You can check for issues with Enhanced Tracking Protection and Total Cookie Protection. Firefox shows a purple shield instead of a gray shield at the left end of the location/address bar in case Enhanced Tracking Protection is blocking content.

• click the shield icon for more detail and possibly disable the protection

You can check the Web Console for relevant-looking messages like about blocked content.

• https://support.mozilla.org/en-US/kb/enhanced-tracking-protection-firefox-desktop
• https://support.mozilla.org/en-US/kb/total-cookie-protection-and-website-breakage-faq
• https://support.mozilla.org/en-US/kb/trackers-and-scripts-firefox-blocks-enhanced-track

Sorry, I'm not sure I understand... proxy settings? Themes? What could this have to do with my issue?

In the case I may not have given enough information, I've signed into two Microsoft accounts on Firefox: my school account, which I use exclusively for checking my school email on Outlook, and my personal account, which I use for everything else. My problem is whenever I go onto a Microsoft website, it often forgets I was signed in on my personal account, and leaves me either signed out or signs in automatically with my school account. In the cases where I'm signed out, clicking the account button shows me the option to sign into my school account or to another account, and clicking the latter either instantly signs me into my school account or brings me to a new page asking to sign in with either my school account or another account (This is also where I end up if I find myself signed into my school account and select to sign into another account). From here I can choose the appropriate option and when prompted to fill in my email, no autocomplete does it as I might expect, but after inputting it I am never prompted for a password—it signs in instantly.

I always have an Outlook tab open on my school account, and I don't believe I've ever been signed out there.

I tried clearing cookies as you suggested but I'm afraid it didn't work. I'll also note this issue has persisted over full reinstallation of Windows so corrupted local settings are unlikely to be an issue.