ODBC, loosing static IP, alternative

My present host requires that I connect to database with static IP. Also for attaching SSL to pages (commodo)

But after finally getting high speed cable, I can finally say goodbye to dialup. With the hitch that the new service does not give out static IP's or even ranges of addresses. Each session is at random.

I understand that DNS can be used in place of static, but don't know the how of this. So I need to find a reliable linux host to go this route, also with uptime guarantee and shell access.

Suggestions? (No virtual hosts.)