I can’t answer your question, but I felt it worth chiming in to say that I have lots and lots of sites on different GoDaddy plans (shared and dedicated), and none of them give me any issues with forms or with sending emails. In the past I’ve used the built-in Rapidweaver forms, FormLoom 1 and 2, FormSnap, DateSnap and MapSnap - all of them have worked flawlessly with my GoDaddy-hosted sites. I’ve since switched to MachForm for all of my forms needs - it’s much more versatile and there’s zero fiddling with databases (after initially setting it up…which MachForm will do for FREE, if creating a db isn’t your thing).
It looks to me like your sites are hosted on an Apache server - I mention this because GoDaddy also offers plans hosted on IIS. The latter will probably be problematic, but you should be fine since you’re either using Classic Linux or C-Panel.