Would severe dry eye cause a droopy lid?

As we age, our eyelids naturally sag and droop. A combination approach would be best to see visible improvement. In terms of non-surgical treatment options, I would recommend a Chemical Brow Lift (using Botox, Dysport, Xeomin, or Jeuveau) and filler injections to rejuvenate the temples, eyebrow, eyelid, eye troughs, upper eye hollows, and forehead. I typically combine this with Ultherapy and thread lifting to lift the brow; RF and skin tightening laser treatments to stimulate collagen production and rejuvenate the skin; and laser or superficial RF devices for improvements in texture, color, and any crepey skin around the eyes as needed. In some cases, surgery is the best solution to achieve the desired results. See an expert for a formal consultation. Best, Dr. Emer.​

I also believe that you need to get your dry eye problem under control before considering surgery to correct your droopy upper eyelids.  But there is definitely help available.  Your puncta (tops of the tear ducts) could be closed with very minor office surgery without the need to have plugs.  Under the care of an oculoplastic surgeon, you might also get a prescription for temporary topical steroid drops to alleviate the burning with Xiidra.  Good luck!

 You have been counseled correctly. You really need to get your dry eyes corrected before you work on your droopy lid. Raising the lid will allow more of your cornea to be exposed to the elements leading to faster evaporation of your tears, worsening your dry eyes.