When I’m in the middle of investigating and I run out of green and gold badges and I can’t get..

Don't start an investigation with less than 300 badges

If you can, avoid that situation by playing when your teammates are most active and can hep give you more time. If that is not possible, you next recourse is to pay diamonds to continue. If you are unable to do that, you will be locked out from playing further, and a a teammate will have to spend even more diamonds to release you. If no teammates steps up, then you cannot play any further in the current event.