I am so sorry this PEG thing seems to be such an uphill battle but I think, down the road, it will be something very positive for you. My Bill still says that it was a Godsend.
You might try this, since you are just keeping the tube flushed. Take the plunger out of that syringe (put it aside) and just pour the water into it and let gravity work. You can change the position of the tube higher or lower to control the flow. That might be less icky to you. Same thing for the supplement, when you start it...just pour it a few ounces at a time of water, then the can and let it slowly flow in. Then pour a few more ounces of water to clear the supplement. Its a great way to get more water when you are in the throes of treatment and need fluids. Cleaning the syringe will be easier without the plunger as well.
Good luck...many of us here are comfortable with this whole process...keep posting and we will help you.