Quits at idle in neutral

I have a 2004 Yamaha 60. Recently it started quitting whenever I am in neutral at idle. It will start good at idle and run for about 20-30 seconds then quit. If I increase throttle in neutral it will stay running, but if I increase the idle adjustment (I had it up to 1200 RPM) it will still quit when I shift to neutral. Yet if I throttle to 1200 RPM in neutral it will stay running.
It runs fine in gear even at low speed and starts easily in neutral at idle.
I've checked the fuel filter and did the link adjustment on the carburetors (I have the service manual) and have played with the idle adjustment screw.
I've also replaced the spark plugs.

You set the idle speed with the engine IN THE WATER right? (not on land running on earmuffs). Idle set on land running on earmuffs will seldom let the engine idle when it is in the water (exhaust back-pressure). It may start in the water but it will seldom keep running (learned this the hard way like everything else).

Yes the boat is in the water. All my observations and adjustments are with the boat in the water.

I'd buy 2-3 cans of jet spray carb cleaner and pull the carbs, break them down and give them a good cleaning... Blast carb cleaner through every hole and orifice you can find. Carb cleaner is CHEAP so use ALOT of it. Remember to back out the brass fuel jets and make sure the orifices in them are perfectly clean and clear. Seems like everything needs pulled and cleaned on a regular basis if you are burning ethanol-blended gas (whether you are adding a fuel stabilizer or not). That's where I'd start as it is not difficult and seems to be the root of many idling/running problems. Don't listen to anyone who says you need to buy new gaskets and such when you pull the carbs. I've pulled carbs over and over again and used all the same gaskets without a single issue. At the least, you'll have eliminated one of the most common causes of such problems.

I did the carburetor clean up as suggested. It appears to have solved the idle problem. What convinced me was a simple test. This motor uses an enriching pump and an exothermal valve instead of a choke. There is a manual valve on the pump which is normally in the automatic position. This position provides additional gas when the engine starts cold and slowly reduces the gas over a few minutes. If I switch this to the open position (this provides additional gas all the time) the motor would run without stalling. This suggests the problem was inadequate gas flow at idle.

Glat to hear you're up and running. If at all possible, ALWAYS run 100% gasoline that is usually available at marinas and a few select local gas stations (ask around and you can search for them on line that sell 100% gas) and run the Marine Green Color Sta-Bil ALL THE TIME (not just for storage). After wrenching my way in and out of my outboard enough times with probs caused by that damned ethanol I'll never put in my boat again. It costs a a bit more but who wants to rig all the rods, get all the snacks and subs made, pack up all the stuff and kids only to have an issue and another day of wrenching on engine eating that sub sandwich that was meant for a great day on the water during a break from the wrenching? I bet 3/4 of the no idle/rough running/no power problems reported here are caused by fuel issues from running gas with 10% ethanol from gas stations. It took me several times in and out of my engine fixing probs caused by that ethanol crap because I was cheap and my engine was rated for 10% ethanol si I figured no problem. Works fine in a car that gets driven every day... absolute disaster in in boats that don't.