CRJ-900 flight crew reported a malfunction with the Stall Warning system in cruise flight.

Narrative

At FL340 en-route to ZZZ1; level flight; constant airspeed; the Stall Fail caution; Stick Shaker Wobbler aural warning came on instantaneously with the autopilot disengaging. Evaluating the situation and crosschecking instruments; I noticed unchanged airspeed; altitude; engine power. With the wobbler indicating an imminent stick pusher; I disarmed my stall pusher and prompted the First Officer (FO) to do the same. The FO was hand-flying the aircraft at that point; not deviating from the altitude. I also noted on my airspeed band high and low-speed Barber poles overlapping. I asked ATC for a block altitude. I then transferred the radios to my FO and ran the QRH for Stall Fail confirming the deselection of the Stall; Stick Pusher. The Stick Shaker contented with an occasional wobbler. The AutoPilot did not reengage. ATC accommodated our requests promptly. I elected to return back to ZZZ. I briefed the Flight Attendants (FA) on the situation and they briefed the passengers; with the note that later; workload permitting I will address the passengers again. Additionally I tried to contact Dispatch/Maintenance via AIRINC multiple times; to get guidance to immobilize the continuous stick-shaker. I also looked through the QRH again; not finding a procedure to immobilize the continuous stick-shaker. The AIRINC connection was garbled and unreadable. I tried to contact dispatch via ACARS to get a different AIRINC Frequency and possibly a circuit breaker location to disable the stick-shaker with no results. Dispatch sent a VOR frequency; a ZZZ Center frequency; and eventually the AIRINC frequency that we tried on the initial contact. Additionally I received multiple ACARS. ATC at some point asked; and I explained our situation. Since maneuvering of the aircraft and its performance as well as; all indications including airspeed were working; I did not request priority handling. My First Officer did an excellent job keeping the aircraft within altitude; airspeed and heading; even though we encountered moderate turbulence at some point. ATC accommodated all our requests and requesting priority handling at that point would have not changed the course of action that we were taking. I did keep in mind that the option of requesting priority handling is available if needed and would not hesitate to do so. I also briefed the passengers via the PA and communicated all necessary information to the Flight Attendants. At around 16;000 feet; the stick shaker stopped intermittently; with the occasional wobbler; and eventually subsided at around 8;000 feet. We landed without an incident.At the gate I deplaned the passengers; making additional PA's passing on information. After that I contacted maintenance to make a write up and explained the issue. Maintenance personal entered the aircraft during the call and I explained again in detail what the issue was. During deplaning and with maintenance onboard crew scheduling contacted me multiple times with schedule changes; even thought I indicated during the first conversation that I will call them back after completing this flight. After a self evaluation and looking back at the workload that we encountered I decided not to take an additional flight that day. At one of the calls from scheduling I asked to be connected to the Chief Pilot and discuss this with him. On a subsequent call I was connected to the Chief Pilot and after a short conversation was placed on hold for that day.After the flight I had discussions with other pilots and an instructor/pilot Training department. I reevaluated my decision to not request priority handling and came to the conclusion that; even though requesting priority handling would have not changed the outcome or workload in this situation; It would have been a better choice. In the future I will take this into consideration. During the flight I attempted to contact dispatch via frequency. The communication was garbled and unreadable. I ACARS dispatch and ask for an alternate AIRINC frequency. Dispatch sent multiple ATC Center frequencies and a VOR frequencies. The procedure for AIRINC should be reviewed with dispatch as well as the radio issues with AIRINC. To my knowledge there is no procedure in the QRH that addresses immobilizing an erroneous stick shaker; I tried to resolve this by involving a maintenance controller via AIRINC or ACARS with no results from dispatch. A standardization in abbreviated texts or phrases to use on ACARS between dispatch and pilots might have solved this. After the flight the workload was still high; deplaning; answering questions from passengers; making a maintenance write up and explaining to maintenance personal that come on board. During this time I received multiple calls. Some were from dispatch; which I felt were necessary; but also from crew scheduling which addressed schedule issues later in the day. On the initial call from scheduling I explained that I was busy and will call them back. I received multiple; additional calls from them; which became distracting.