Is there a way to add multiple modules to a game?

I am new to Call of Cthulhu so I've been using modules, and wanted to add the module paper chase to my game, and it looks like I have to make a whole new game. Is there a way to just add it I don't know about or is it impossible?

Not easily, no. Modules are usually large enough that if you could add them to existing games, they would become unmanageable, leading to lag, inability to make backups, or recover in case of corruption. Normally CoC modules are not that large, but decisions are made by the creator or converter of the game. Addons are usually for smaller adventures, since they consist of a handful of maps and handouts, in a self contained folder that can be deleted after the game is played.

A Pro subscription does allow you to use the Transmogrifier, which makes it easier to move assets between games.

I am assuming you also cannot simply remove a module from your game to replace it with a different one? 

No, modules merely provide initial content in a game. There isn't a way to identify that content as coming from the module after the fact. 

In case it helps... We have multiple games that we bounce back and forth to. The players simply use the Character Vault to transfer from one to another. 
We have found that this also helps to not overload a single game resulting in lag.