The neti pot is a really good idea.

Dissolved gases in the water can make the water feel more irritating to your sinuses. Usually this is just air, but keep in mind that tap water tends to be chlorinated. If you bring your water to a boil and let it cool before using it, you can use it while it is still just a little warm and it will feel better.

To 1 quart of water you can add 1 teaspoon of salt, to make your own instead of using packets. With your sinuses being irritated, you can also add 1 teaspoon of baking soda. It is better to use the solution up entirely rather than making some and saving it for later. You may benefit more from boiling 1 cup of water and using 1/4 teaspoons of salt (and baking soda) and doing it 4 times if you do not have time for the entire quart at once. Be sure not to rush it, because it can go down the back of your throat and come out your mouth if you pour so quickly that it can't get out the other nostril as fast as you are pouring it in. Neti takes time, but it works for me even when advil and excedrin fail completely.

When you are done, just rinse your neti pot and let it air dry.

You can do neti leaning over a sink, but some people prefer to do it in the shower. It isn't that messy, but you are talking about washing the insides of your sinuses and some people think that is a bit too gross to do in the sink. Personally I don't think it is any worse than raw chicken, but that is just my opinion. I just use the sink and then clean up the area afterward the same as if I had been handling raw chicken, to avoid leaving bacteria or anything else around.

And you can probably get videos on YouTube of someone doing neti if you are not sure about how to approach this. There are even supposed to be ways of doing neti without an actual neti pot, but I didn't need to look them up because I already had a neti pot. I have heard of people using styrofoam cups and pinching the side to make a sort of spout, but I can't imagine how that would get the solution into the nose instead of having it run down the face. So I would suggest looking for those videos.

One more thing. If you boil the water, add the baking soda after it cools to the point where you can pick it up comfortably with your hands. When baking soda gets hot, it breaks down to give off carbon dioxide. This is what you were trying to avoid by bringing the water to a boil.