Webmail on simple managed / shared host.

After an interruption in progress on another large host, I started the migration to a new host. My 2 previous hosts were based on Windows (things like MSSQL are needed), but I disconnect the main site until we review everything. So I've moved to a basic shared host. All I need right now is to access a few simple pages in about 5 different domains, along with a reliable email. I'm just using any installed application, but I could do it in the future. There is no CMS, although I can search it in 2 different domains.

However, now I have a problem with CPanel that I wonder is inherent in this system. In previous hosts, I had no problems to remain connected to the control panel (usually Plesk) or any webmail. I'm in several networks (usually from home, office and mobile) and these networks change the public IP, something I can not change, since this is at the corporate level (most of my sites and emails are basically personal). The email will not remain connected in Roundcube for more than approximately 2 minutes in the corporate network and 5 minutes in the home network. The control panel of the host runs out quickly: I completely lost the tickets that I wrote, only at the moment of sending them so that they are lost when I return to a login screen.

My current host is telling me that there is nothing that can be done about it since it is CPanel. Is this true? Is it better to go back to a Windows host? I need an email based on the web, since I am using approximately 5 different devices.

Any other idea?

WHT looks great, I wish I had found this a few months ago or earlier!