Having an IFR flight plan means absolutely nothing once you go into class G. Once you leave controlled airspace you are on your own until someone clears you back into controlled.
Commercially we handle it by.
1. Get speaking to the next controller on box 2 5-10 mins before you get anywhere near the controlled airspace.
2. Dump who every your getting a service from when you want to and don't be bullied into staying with them. Its class G ignore their bleating if they want to keep you and be a Captain.
3. Have a escape route ie never put your aircraft somewhere that you can't escape without busting airspace. Be very careful choosing left or right to turn away.
Personally I don't take a service off these approach units anyway in class G because they are just a pain in the backside these days and don't realistically offer you any real safety benefit with the rubbish level of service that they are willing to provide. If they give you more its because it is for their benefit not yours so they can move you out the way of their own traffic.
So basically grow a pair and be a Captain control the situation, they are there to provide you a service not the other way round. If they are not helpful don't speak to them again and then they have the problem of trying to fit a 737 under a de-confliction service though multiple unknown traffic paints.