Are you still having problems?
A while back I had troubles logging on to an email account, I could do so from one computer but not the other. Troubleshooting the problem I came to notice I had a 32bit version of Firefox running on my 64bit computer. Don't know if that contributed to the problem but changed out the version as soon as I found out.
Next step was to start Firefox in Safe Mode and attempt a login. FireFox Safe Mode disables all the extension loaded on the browser.
There's a number of ways to find out what extension you have load. To me, the easiest and most comprehensive listings is to click on Help > Troubleshooting information, when that window opens scroll down to Extensions.
In my case, it appeared there was an extension causing the problem but again not sure if FF version had any bearing.
I believe Palemoon is a clone of FF, if so you can find Safe Mode by clicking on Help on the top Toolbar. Then click on Restart with Add-ons Disabled. A popup will open asking you if you want to restart, if you click Restart your browser will close and reopen.
I googled Palemoon downloads for a 64bit OS (operating system),
here's the link. Reading the page info it states unless you have a dedicated GPU (graphic/video card) you should use the 32bit download version of PM regardless if your OS is 64bits. Your newly acquired laptop is most likely a 64bit but it's unlikely to have a video card, rather running embedded video off the motherboard.
On the Desktop, right click on This PC icon and then click on Properties to find out which OS you're running.
To find out what version of PM you're running click on Help and then About Palemoon. If you're running a 64bit version I would expect it to say 64bit after the version. If you're running a 32bit I would expect the absence of both 32 and 64bit next to the version to indicate the version as 32bit.
If you read this far and if you were able to login from Safe Mode you will now need to disable you extensions one-by-one until you find the culprit causing the problem. You do this by clicking on Tools > Add-on > Extensions and then disabling them one-by-one. After you disable an extension you'll need to close and restart your browser in order to clear you cache.
To ensure you cache is being cleared when you close your browser go to Tools > Options > Privacy & Security and scroll down to History. In that section tick the last box "Clear history when 'PM' closes" and then click on Settings and be sure Cookies and Cache is ticked off in their box.
Sorry this is so wordy........sorry for your troubles.....Good Luck...!!