The valve isn't shutting off all the way. That's why it continues to run. Low flow is probably the reason. That's why it shuts off when you remove an emitter.
If the valve has a flow control turn it down as far as you can. Don't turn it down too far or it wont function.
If you can't get it to work you'll need a low flow valve. I'm not going to recommend one. Someone else might have a favorite valve for drip. I don't.
Here's my favorite drip emitter for pots. I've never had a complaint from a home owner. Plus they put out enough water to shut off the valve.
Full Circle
Half Circle
There's also a full circle spray which is cool and a flood bubbler.