My question is if I get the client to sign up for a DO account, then they hand me the credentials. Am I able to hook this account up to my forge account (which is already linked to my personal DO account). So that I am then able to create and manage servers for the client?
Forge needs ssh key to access DO instance and setup things. So, yes.
I've seen elsewhere about using the server providers section for access to other DO accounts, but wanted to ensure that this was the case for my Forge subscription type of "growth".
Yes, you need at least "Growth" plan to be able to manage multiple servers.
How do others who manage clients in this way deal with things on DO & Forge?
You said "who manage clients in this way", so I do not understand the question. If they deal it in that way, then what do you mean by "deal with things on DO & Forge"?