Well it would be handy to know what mac model and year your mac is. The age of the mac will determine
which OS you can upgrade to.
Open Safari and click on this link, How to get old versions of macOS – Apple Support
Go to the Check Compatibility section to see which OS or OS's you can upgrade to.
Then go to the Download OS section and click on the OS you want to install.
13GBS is not very much space at all, you should free up more space.
In general you should keep between 15-20GBs free on your drive at all times,
if you operate your mac with very little free space it could lock up and become
completely unusable.
If your mac can upgrade to Big Sur, you would need much more free space.
The Big Sur installer itself is 12.2GBs in size so once you downloaded that you
would have next to no room left on your mac to install it.
How to upgrade to macOS Big Sur - Apple Support
In the first section Check Compatibility it states that if you are upgrading from an
OS earlier than macOS Sierra you need 44.5GBs of are space.
Before upgrading make a full back up of your Mac.
Check that third party apps you use have updates available for the new OS.
https://roaringapps.com
Check that your printers/ scanners etc have updated drivers and software available for the new OS.
Connect your mac to your router via cable rather than rely on WiFi.
Do not use your mac while it is downloading
and do not use any other devices that share the same internet connection.
If the App Store says ‘Open’ then it is detecting a previous download of the install app
go to the Applications folder locate Install macOS ******.app.
Move it to the Trash, empty the Trash.
Restart, click on the link again, the App Store should say ‘Get' or 'Download'.
If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.
Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,
some updates may be free, some may be paid. Use this app to discover which apps you have
are 32bit, https://www.stclairsoft.com/Go64/. Once the list has been created you can then
contact the developers of the apps to see if they provide 64bit updates.