Is there a shifter solenoid that could have failed?

Sounds like perhaps the battery has lost needed power -- that's one idea. Have the battery tested first for state of charge and the ability to sustain a load. Since you've been trying to start this engine a lot, it could be that it's run the power down. Beyond that, sure, a transmission shift lever issue could be in play -- but I highly doubt it; I think what you are experiencing is the effect from a different cause...such as battery power issues. After you sort out the power issues, you'll need to get into engine performance and find out why the vehicle won't start! Don't fixate on "fuel" -- it could be ignition spark, air to the intake manifold, engine compression, engine timing related. Hands on testing is needed. www.repairpal.com/repair-shops