Have a 19.5 Briggs in a John Deere . Have replaced starter twice ,& solenoid and battery . After running it won’t start back . Just barely bumps the flywheel . Tonight I adjusted the valves to factory specs hoping that would help. It started first try ,& then 5 min later started doing the exact same thing . I’m at my Witt’s and with this thing Help !!!

Might want to check the compression release.
 
Looking under the valve cover, the exhaust valve SHOULD have a slight bump when near the compression 
cycle.
 
Also, make sure you don't have a leaky carberator leaking fuel into the cylinder
Fluid doesn't compress and would hamper/stop cranking..
 
 
 
. 

Well , pretty sure at this point it’s definitely the decompression release mechanism that’s shot . Starts cold just fine , I mowed tonight with it . Went to start it back up like a minute after and just got a little whine sound , so I stopped and got off . Turned the top of the motor counter clockwise until it got a little tough to turn , it started right up . So I guess my question now is , I’ve had several quotes on this repair ,& the lowest I’ve got was $600 which is what I paid for the mower . It’s a 2009 La105 model . I don’t honestly feel confident I can do this job even with the YouTube vids I’ve seen . I’m not sure to just keep doing what I’m doing and hope it doesn’t mess up the engine at some point , or bite the bullet and have it fixed . What would you do in my shoes ???