If I am trying add a parameter to multiple scripts, do I need to create new parameters with distinct IDs for each? I don't see an option to add or associate existing parameters, and if I try to add a new one with the same properties, I get an error saying that parameter exists.

prefix your param id with something related to the script. eg for Consolidated Invoicing, prefix the param with ci whilst for Advanced Payment, used ap that way you can have the same script params but for different use cases

gotcha - that is what I thought I might have to do, but wanted to be sure I wasn't missing something. Thanks!

Nah no special tricks 🙂

If you need the same value usable my multiple scripts, you can just create it on one script and make it a company preference. Then all scripts can access it.