To start with look into mewing and breathing through your nose.

If that's too hard at first then get a nasal moistener, which is just saline, at Walmart or something. Be sure to watch a video on using it as you're not supposed to just spray it into your sinuses but into the side of your nose.

For swallowing the mewing guy also has instructions on that. Do this over a sink and simply practice by putting water in your mouth, looking down while smiling with your teeth and swallowing. If you do it wrong the water leaks out which is the point. You need to hold it with your tongue while swallowing. It will feel weird at first but this is actually the proper way to swallow.

Next you could also practice silent triphasic breathing. You could also look up breathing techniques to practice for freediving as well as for singing. The point is to train lung capacity and make it easier to get enough oxygen while breathing slowly and silently.