Hi TR, you just used the magic word: gulp. there has been much discussion here about the fact that one of the hardest things for people to do is to learn to sip water or sip any beverage rather than gulping it. You can't gulp water any more than you can shovel down your food really fast. You'll get sick. But I wonder if you might have better luck with water drinking if you used a teaspoon and put your water in a short glass and just spooned it into your mouth one teaspoon at a time. You could pretend you're eating soup, really really really weak soup!
Nicelady's comment about using stool softeners is a good one if you are able to get the urge to go but you just can't make anything come out. I never had constipation in my entire life until after bariatric surgery. The combination of the re-plumbing and he opiate-based painkillers caused me to be constipated like I never would have imagined you could be! Stool softeners saved my life.
Before I tried milk of magnesia I'd be taking a laxative like colace because I'd be puking that stuff up and I don't think that it works really well.
It is really important that you get that constipation under control immediately. All kinds of bad things can happen if you don't. But you can't take anything that is going to mess with your new plumbing.