First of all there is not a 'thing' called an ADO. Do you mean 'multiple ADO Connection objects to connect to Oracle?'
I believe you can change the number of concurrent sessions that Oracle will permit.
Do you really need the isolation that the setup you are using creates? Could you have one global connection within the application, which you then use to open multiple recordsets? I presume someone logs into this application, and that the application then contacts Oracle for the necessary data, using that username and password.
If you are using the login of the user of the application for each connection (if 'connection' is what you actually mean), then one connection would work as well. If the username and password are different for each connection (if that is what you mean), then you do need to use independent connections.
If the user logs in, and you use their login for some things, but the application also has its own internally stored username and pasword for other things, then you would need 2 connections. But that’s not all that many, compared to what you seem to be saying is the current situation.