Need help with a question

I think your thought process is spot on. The first thing that came to my mind reading your post was a possible stroke. Is the patient on any anticoagulation?

Hard to assess if the BP is the culprit here without knowing the patient's baseline, but you're right to assume that it could be especially with the dilt gtt. Has the drip been running or is this recently started?

Hopefully BG's are being monitored with the pt's hx of DM and NPO status.

Agree with OP and PP: suspect stroke, meaning call RR...in the meantime, get a blood sugar.

Although not directly r/t stroke, would stop the dilt drip, the appropriate action in the face of low BP ~ until alternative rate control measures can be implemented.

I would definitely think r/o acute CVA and checking BGL would be the priority here. In addition to what everyone else has mentioned, what is the pt's fluid volume status like? Is pt on blood thinners? Any reason to suspect bleeding somewhere, which could account for lower BP and compensatory tachycardia. Does the pt need any additional IV fluids - (fluid volume deficit)?

Also, could there possibly be some infectious component - possible sepsis? Pt has COPD, maybe PNA? What's the temp, labs? Hypotension, tachycardia, AMS/hard to awaken could all possibly be due to sepsis.

Your process seems right to me.

I don't think you're missing anything. Blood sugar check is part of our stroke protocol. Could be the TIA or a stroke- could be very likely with afib, esp if new afib and not anticoagulated. Could be going into failure also. Would probably stop the drip- hopefully getting verbal order with provider you are stat calling.

I'd call for help , see what they advise right now. 

Here’s what I would do in this order:

1. Stop gtt and check the rhythm patient is in, did they ever convert to sinus? I’m assuming I’m in the ICU where I work. 
2. Quick neuro check, arms, speech, pupils, orientation if able.
3. Glucose check. (Stop here if low BG and treat)
4. Call resource or charge RN, let them know I’m calling a calling a code stroke.
5. While waiting on others, orders, etc, get ready to leave for stat ct. 

All of that can get done in 5 min. Next thoughts are that yes, patient could be in failure. If you are able get a listen on their heart and lungs, any obvious murmurs or crackles? Jugular vein distention? Ruling out stroke is the priority because you have a window for treating an ischemic stroke, so I would have done what you suggested for sure.

What you describe is how I normally wake up, so I wouldn't get too far ahead of your skis just based on that.  Is the speech slurred due to unilateral facial paralysis? is it dysarthria or just mumbling?  Does it persist once allowed to actually get their wits about them?  

87/60 is not a critically low BP, and not necessarily concerning in a sleeping patient.  Decreasing the dilt gtt may improve the BP but keep in mind it might also make it worse depending on how much the HR increases.