Taping your mouth may or may not work. I would look into your nose first, do you have any nasal congestion?

A little bit because of allergies

That's your main issue right there. If you are not taking anything for your allergies consider having a doctor prescribe for you nasal sprays and anti histamine for one month to see if this makes a difference. Also, if your turbinates are inflames that can help.

After one month if your nasal passages clear and you are still waking up with open dry mouth you might want to look into taping your mouth.

Tape mouth, and buy nasal dilators like these. I wouldn't recommend that exact model necessarily just linked it so you know what I'm talking about. I just bought these and I'll let you know how they end up working.

I personally found out how useful these could be the other day. You see in my particular case, and I imagine a lot of people with CFD's cases, my nasal passageways are more restricted than they should be because of the way that the skin has tightly grown around the nose. I have a slightly hooked nose at the end, and as a result it obstructs some of the nasal passageway. I can still breathe fine when I'm at rest, but it's almost impossible for me to run and exercise while only breathing through my nose because of this, and that's very important. You can try yourself to see if you'd benefit. Just manually take your hands and push upwards on your nose to make it look like the people of Whoville, but with longer nostrils, if that makes sense. If you have a noticeable difference in your ability to breathe through your nose, then you should try these dilators. 

Yes, the nasal dilators have helped me. I've just used the Mute brand so far, but I plan on going thru the list, eventually, ordering off of Amazon.

I self diagnosed as UARS - almost all my symptoms wrapped in in one name! - and that brought me to this website. My symptoms were so severe I self-diagnosed as Raynauds phenomenon 2 decades ago (On the Raynaud's diet you abstain from chocolate and caffeine, and restrict/limit many other foods!).

What's helped?

ABC chiro care with endonasal cranial correction and nasal dilators. I taught myself to lipseal years ago, and had my teeth expanded years ago with invisalign.

It's not perfect, but it helps.  

And yes, just like @zosogg manually pushing up on my nose helps improve my airflow a lot.
Zosogg, have you found anything else that's helped you in your journey? Did you notice a change in any symptoms or energy level?