Start with the simple stuff. Replace the spark plug. I have had several in the last few years act like they were running out of gas and it was the darn spark plug breaking down when hot.
The first was on my JD 14SB hand mower. It would run great for 5-10 minutes and then sputter an die. Just like it was running out of fuel. I had the carb off 5-6 times. I even removed the complete fuel system and cleaned it an replaced all the fuel lines. Son suggested trying the spark plug. I told him it fire bright blue when it was out. I tried it and sure enough that solved the problem.
Had my Honda Powered pressure washer acting the same way a few months later. Changed the spark plug and it ran great.
So try replacing the plug. Your only talking a few dollars to try that.