I wish to make a complaint about your services and servers.
I want to be able to use wordpress but your servers return an error when trying to upgrade.
I called for support and that was a nightmare because the customer service rep blamed the problem on me for using the 'linux ultimate' package that was old even though I am paying full price for the service I am using.
Godaddy was good a couple of years ago but now it is just overpriced garbage.
If I cannot use wordpress on my service then I will be forced to legally take my problem to the ministry of fair trading (the courts) in Australia.
Did you ever try how to install manually through uploading files via FTP account? If it didn't work for you then yes you need to change your hosting plan.
Because the hosting you might using is too old design and WordPress is keep getting updates requiring recent PHP version and sometimes memory handling which might not be supported by your current host design.