Practical Domain Change Question

Hi guys, I have read a ton of support articles on changing my wordpress domain, but none seem to answer my main question, so let me try to explain.

1) I now own 2 domains from NameCheap (let’s call them currentdomain.com and newdomain.com)
2) My host is Hostgator, and my plan does NOTallow me to have multiple/addon domains.
3) My WordPress installation is currently on currentdomain.com

I want to move my WordPress install to newdomain.com, and re-purpose currentdomain.com for another venture. My fear is that when I switch my primary domain with Hostgator, I will either lose my WordPress install or not be able to navigate to my login site. Or am I not thinking about this correctly?

Is what I want to do even possible without add-on domain capability from HostGator? Basically its that lag time during the change that has me concerned.

In the alternative, is it possible to back up my WordPress site, change my domain, install WordPress on the new domain, and then load my backup into it? I don’t care about downtime in the least.

Thanks a lot in advance.

Andrew