If your droplet is full, like 100% full it might cause issues with other services unable to write data. Usually, this is caused by a plugin of your website/application or anything similar not being configured properly.
What I’ll suggest you do is SSH to your droplet and start digging around. When I need to check for disk usage, I check for big files first. You can do that with a find command:
find / -type f -size +100M
Wait for the command to finish, it might take a while!
Next if you don’t see anything that can be removed, would be to check folders with the biggest number of files. Let’s say sessions, your application can generate session files (like Magento does) and after that not remove them. They can be small files but if they are too many, the disk usage will still be up.
You can use the following command to find the directories with the most files inside:
find / -xdev -printf '%h\n' | sort | uniq -c | sort -k 1 -n;
If uncertain what can be removed, I’ll suggest contacting the person that developed your website, they should know what can be removed or not.
One last thing, if all files are needed, maybe you just need to upgrade your droplet to have more DISK space.