Can I do multiple neutrals and potentially make a bad situation worse?
You can add another to the connection of existing neutrals, you will likely need to change the yellow connectors on the neutral and ground with larger (red) connectors. Most yellow connectors are only NRTL Listed for 3@#12 or 4@#14.
The existing switch should also have a connection to the grounds, you will need to make sure you connect to the ground of your new switch.