The model number is 287707-0224-01

***"I tried to turn the "wheel" on top of the engine but it looks like it turns one or two full times before it gets tight again."***
What you describe in the sentence above is normal and is not a symptom of any defect.
The description of how it only wanted to run when the choke was applied points to a carburetor needing a thorough cleaning of the internal parts.
Why it quit running altogether when you were mowing with it hopefully just means additional trash in the fuel hoses finally blocked the carb passages completely, or trash coming out of the tank finally clogged the fuel filter.
If this engine is equipped with a fuel pump, it's possible that the fuel pump could have failed also.
In a case of 5 years storage you will need to clean carb and fuel tank.
You also need to replace all the fuel hoses and fuel filter.

Now, the detail about how you hear the starter sound but it looks like something is stalling the engine:
This is a Briggs Over Head Valve type engine (OHV). Briggs says to adjust the valves annually.
It only takes for the valves to be out of adjustment a little bit to cause the compression release feature to become ineffective. So I would think that your new battery has been used to crank the engine much more than it would have if you were not having the other problems...........and you have not been able to run the engine long enough to recharge it............and with the valves needing to be adjusted, the weak new battery just can't spin the engine past the tight spot.

I believe you need to call the man to handle these issues unless you own a fairly good collection of basic mechanic tools and have some basic mechanic skills.
If you feel comfortable attempting a DIY repair to the issues, there are people on this board that can offer guidance, but it is you who would be performing the work.

Thanks for responding.

While it would be easier to call someone, I'm actually interested in learning how to fix this.

When I replaced the fuel filter, I didn't know if I put the filter facing the correct direction. Do you know if that would have caused a problem?

I was able to get the carburetor out and re-install with out much difficulty. The only part that I could not clean was the float bowl. I have a fuel solenoid (?) under the carburetor and I didnt know how to get it off. I didn't see any screws.

On a scale of 1 to 10, 10 being very hard, how difficult is it to adjust the valves?

Thanks.

The fuel solenoid actually just unthreads in the same fashion as a typical bolt, there should be a "wrench flat" built in to the solenoid body. The fuel solenoid is what secures the bowl to the carb casting. And.........inside the bowl in the same bore that the fuel solenoid threads into, is the main fuel jet.......and that is what needs cleaning out the most.
So, if you haven't gotten to that part, you really have not affected a good carb clean out.
If you have never adjusted valves......ever in your life.....on any engine, you may find it somewhat daunting.
But everybody had to start somewhere.
You can e-mail me for my treatise on the Briggs OHV engine valve adjustment procedure. Use the forum e-mail provision.

OP - fuel filter may have an arrow stamped on it indicating flow direction..

I think I still want to try and give it a shot before calling someone for help. Thanks Mownie & rcbe for your tips so far.