When you get disconnected, Look at the westell modem and i believe if themiddle light blinks. It sounds like you are having syncing issues with verizons servers. Call up verizon and keep pressing the issue.
Dont think its possible to have "always on" connection, but you seem to have another issue.
i have verizon dsl and i had the problem of getting disconnecting every 30mins. called verizon a few times, finally speaking with someone in the engineer dept and about a week later, its fixed and its been rock steady since.
also goto the
www.dslreports.com. their forums are very helpful.