output cache will indeed cache the site as a whole. This said however, background tasks arent running on your site, as stated by the banner there. You need to address that, as it may be that items are not being correctly cleared. You may need to switch to cron to run these, which would be our recommendation
Having a limitation on mysql is very much only done on cheap hosting though. To be honest, you should find hosting where mysql is not limited in that manner. 8gb is not a large size for databases on sites these days. Have you considered a cloud hosted environment, then you dont need to think about your hosting at all?