I’m on supplemental oxygen 24/7 and am being annoyed with some nasal congestion and irritation, especially at night. If you have had this problem how did you deal with it. My doctor shook his head in acknowledgement and responded with something like “mummm?” and left the room.
With CPAP (which isn’t quite the same thing), you can get masks that go over the nose and mouth (and even in some cases the whole face) to solve the same problem.
I haven’t tried it, but my CPAP technician mentioned that if my nose became irritated I could ask for a product called *Rhinaris *at the pharmacy. Apparently this is some kind of nose lubricant.
Saline gel spray (like Rhinaris) is great for inside-the-nose irritation and dryness. I’m assuming your O2 is humidified - that there’s a little bubbler of water on the machine. If not, talk to your doctor and see if you can get humidity added.
If it’s the outside of your nose being irritated from the tubing, you can use a little lotion to help lubricate the skin. You can tear and stretch a cotton ball and use it as padding on the pinchy spots, or you can put a bit of tegaderm film on the ouchy spot to reduce friction - that works better on the cheeks than the nose, due to the shape of the nostrils. But what’s mostly going to help is time for your skin to adjust.
For the runniness, you might talk to your doctor about using a saline neti-pot. I taught a patient to use one last week, and it, and I quote, “changed my m----f—ing LIFE!” 
(I had to give her a high five - I’ve never seen anyone else do the entire 8 ounces of a neti-pot their first time. She was a trooper. And afterward, you would have thought she was high, she was so delighted.)
Flonase (a corticosteroid nasal spray) and Singular/ montelukast (a leukotrine inhibitor taken by mouth in pill form) are medications I often see people taking when they’re using oxygen. They may help to reduce nasal congestion in some cases by reducing inflammation. They are not without risks, of course, and that’s a conversation for the doctor, as they’re both prescriptions. I have no idea if they would be right for you or not.
I second the Neti pott suggestion. You basically are flushing out all that mucus with saltwater. Safe and it directly addresses the problem (versus taking drugs that reduce congestion, which do many things). I do wonder why there’s congestion - what about the oxygen creates the irritation? - but that’s a physiology question that wouldn’t be helpful for fixing it.