Wow, it's hard to know where to start on this one, but here are a few things to check BEFORE you start replacing a carb.
1. Is the air filter clean?
2. If there's a fuel shutoff valve in the fuel line, is it turned on?
3. Have you replace the fuel filter WITH THE ONE SPECIFIED BY THE ENGINE MANUFACTURER?
4. Is the fuel hose clear?
5. Is there gunk in the fuel tank?
6. Have you cleaned out the small fuel reservoir on the bottom of the carb?
7. Does the float in the reservoir move freely?
8. Did you leave fuel in the carb and/or gas tank over winter? If so, you probably need to do a thorough cleaning (and this does not mean merely spraying carb cleaner into it - it means disassembly, soaking, blowing it out with compressed air, and running tiny wire through some openings that are barely visible.)
9. Have you been twisting the adjustment screws on the carb? I've never known them to un-adjust themselves over a winter season.

As for the fuel solenoid sticking, have you checked electrical power to the solenoid?

If the carb has worked in the past, and you determine that you need a new one, replace it with the same model. However, a thorough cleaning and an inexpensive rebuild kit will normally fix things up, at least as far as the carburetor goes.