when you go to the dentist, the dentist and/or hygenist will usually work with two tools. If they need to rinse your teeth, they'll first blast them with some ice cold water, and then they'll stick a straw-looking thing in your mouth to suck up the water they just skeeted in. Similarly, when they're drilling your teeth, they'll use the little straw fellow to suck up the tooth dust so you don't get weird shit in your lungs.

It's like an oral shop vac, where your mouth is the mess. They use them so they don't have to stop and let you spit. OP has Problems™ and wants to stick one of these things in their nose.

OP, please, just blow your nose or take a decongestant if you're sick (NOT oral phenylephrine because that shit is useless). If you avoid blowing your nose because you get bloody noses all the time you can actually get that part of the inside of your nose cauterized. I used to get at least one nosebleed a day, and it was genuinely life changing for child me when the doctor stuck that swab in my nose. It was painful for a sec or two, but I only get nosebleeds once or twice a year now.

EDIT: netti pots are good, just make sure you practice good hygiene with those. Very hot and steamy showers and putting damp heat on your sinuses is also quite helpful in my experience as a lifelong severe allergy sufferer.