A UDC is a universal data connection, so any data connections in your forms must be converted to a UDC and hosted in SharePoint.
You should not need to make your form full trust. This might be the issue since you would either need to sign it digitally or create an installation for the users.
Hopefully you have no code which should be the only reason to make it full trust, and again I would avoid this complexity if at all possible. 95% of solutions should not require code and can all be done with rules and actions in InfoPath.
Create a new form, place a couple fields on it and create a submit data connection to a library. Leave everything default and publish and validate.