I was wondering how to clone trigger.newmap or how to still have access to the references associated to it when I want to pass this into a trigger handler.
If you're looking to clone the map itself, then the following should work for most needs:
Map<Id, Account> = new Map<Id, Account>(Trigger.newmap);
... where Account is the specific type of the trigger. This will create a new map, such that if it is modified by later code, the original map of Trigger.newmap will remain unchanged.