So I understand you have 2 hosting accounts is that right?
You want to move websites from one location to another?
if so you are going to want to start by backing up your data and moving it to the new hosting account and be sure you backup any databases if your websites are cms based like wordpress, joomla etc...
I don't like the old backend interface and want to 'upgrade' to a cpanel for my current hosting plan which is ultimate hosting
If the hosting plan you currently have does not include cpanel you will then need to migrate the site to the new environment that does include cpanel. The only way to install cpanel is if you are on your own vps which can be complicated.