I can only tell you what I do based on an ENT
It needs to be PVP iodine. Dont ask me why.
I dont use it in my mouth for fear of staining teeth.
Sometimes I use swabs and swab my nose. Sometimes I use a nasal spray I made with saline solution. Sometimes I use a PVP iodine prepackaged swab when I am out of my house. You can buy them from CVS. You can also buy them from Amazon but mixed reviews on them. The ones from CVS are quite big.
Clearly it is better to snort it so you can feel it in the back of your nasal passage but I dont always do that because I cant stand it.
Another solution is something ENT’s tell people to do. Baby shampoo irrigation. I dont do that. I just stick some gentle soap up my nose
BTW, if you are snorting it you are suppose to dilute with saline solution.
The other thing that could possibly work for your mouth is gentian violet. Moms use it on babies for thrush.
HOWEVER it stains. I use it outside. It supposedly can cause cancer. AND I dont’ know if it has been proven to work on COVID
If you swab the nasal area out, or as you say snort it, does it need to be rinsed for fear you will swallow it if it hits the back of your throat? Also does the nasal area need to be rinsed afterward? I’ve only used iodine typically to be absorbed through skin