For wordpress it should just be a case of changing the website address in the configuration before you change the domain name (it will break when you do this but then should be ok once the domain is changed). You may also need to change anything that uses the old domain name such as uploaded images etc.
I don’t know about gallery it may just involve changing the configuration too, I’m sure there will be someone somewhere that has done it.
When you register the new domain just point it to the folder the current site is in.
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