OK, Gotchya!
Sorry to hear you're having problems.
I have more questions than answers at this point so lets get started.
Are you pulling the fuel from a tank IN the hull or from a portable one?
If you are using fuel from an in hull tank, do you park the boat outside?
I think you may see where I am going with this. Because, it sounds as if you may have contaminated fuel. If you are not sure....you NEED to get sure about the quality of the fuel you're feeding the outboard.
Find your carburetor drain screws and THOROUGHLY drain the carbs. Then, hook up FRESH, CLEAN fuel and use the primer bulb to "purge" the fuel lines by leaving the drain screws open and allowing a pint or so to continue come out. Make sure you do this outside and away from sparks or heat sources. A fire extinguisher nearby is always advisable.
If you drain the contents of the carbs into a clear container, you should be able to readily see any water or small debris that may be causing you problems. Just allow the "sample" to sit for a few minutes and if there is a "ball" or more of water at the bottom then you have found your culprit.
Then try and run it again on the CLEAN fuel to see if there is any improvement.
You say the primer bulb will now get hard. But, does it STAY hard or does it start to soften up a bit as the engine runs and then stalls?
Your statement about seeing bubbles in the fuel lines may indicate you have an air leak in one of the connections or in the filter housing. You will need to find that and fix it. It can be a pain since, if the leak is on the SUCTION side of the primer bulb, it won't leak fuel but it WILL pull in air.
I use two methods for finding air leaks....smoke from a smoke machine....something you probably don't have.....and/or REGULATED compressed air and SOAP SOLUTION. I use a shop air compressor with an air pressure regulator set to less than 10 psi. I soak whatever I want to test with dish soap solution and then use a rubber tipped air gun to deliver the air through the fuel line or through a fitting at the filter housing. Of course, you will need to block off the inlet side of the housing or hose to keep the air from escaping into the tank.
You can actually test this way effectively just using LUNG power as long as you aren't trying to fill too large a space with air. For example, you can soap a fuel line, pinch off one end, and then blow into the other and watch for bubbles. But, shop air is much easier and more effective.
If you don't uncover any leaks OUTSIDE of the tank doesn't mean there aren't any leaks. You could have a cracked pickup tube INSIDE the tank that will suck in air and stop the engine. It doesn't take much! That's why it is always good to have a portable tank that's in good shape to pull fuel from as a "reserve" even if you have a large, in hull, fuel tank. The 3 gallon portable is a great troubleshooting tool.
I hope you get some fore Ideas from the guys. Keep us posted on what you try and what you find.
I will be standing by.
Good luck.