Purchasing storage does not expand the storage on your computer. What you need to do is get an external hard drive and go through your computer and either move files to that external drive and/or delete files - the ones hogging the most space are usually photos, videos, movies,etc. You need to maintain a minimum of 10 - 15 GB of empty space at all times for the OS and system to function properly; if you fill it up completely, it will simply stop working.
As for running slow, there may be more to it than not having updated; it could be AV software, the disk being too full, the hard drive failing, or several other things. So, please download and run etrecheck, it is a diagnostic tool. Run it and post the entire report here using the third from the right icon for additional text in the toolbar of the reply window. Then we can take a look.