How's it going ?  Do pain meds not have any effects to allow you to sleep longer.
What have you done / injured in your shoulder ?  
Having that broken sleep is a killer as we're not able to rest & recover properly 
And as you say we end up getting up even more tired/ worn out.
Right leg went out from under me when doing a Muay Thai quick switch kick.   I fell to the right and reached out, resulting in a dislocation.  It felt the same as when I suffered a shoulder separation 20 years ago, so I thought that I'd just get a referral for PT.  Nope.  Next morning I was in agony and went to the ER.  Doc popped it back in - no ketamine. 

 There was a nurse standing by with a syringe of it, though.     
It resulted in the terrible triad: dislocation, torn rotators, and damaged brachial plexus nerves.
Two months in the nerves started waking up.  Nerve pain is no joke and has been as bad as the shoulder joint.  Gabapentin for the win.
Unfortunately, the most painful position is when lying down.  The amount of pain med that would allow me to sleep would be more harmful than beneficial.  Getting up to 5 straight hours of sleep has REALLY improved my energy levels.
I'm having PT to increase the ROM.  Ortho said he won't operate unless I have near-normal ROM.  Getting a second opinion next week - wish me luck!