Yes they stick but you can keep them sort of warm. The mouth is also an issue.
Fingers are the biggest problems. We practiced at 630 so it was cold in Iowa. Marched at Kinnick at half once. Was like ice skaying on the old astro turf. Think is was the 5-3 barnburner with AZ or Utah.
When it is really cold then yes they can freeze, and no blowing warm air through the horn will not completely keep it from freezing - too much tubing. Spray bottles filled with everclear instead of water will work most of the time. We did have one game back in the 90s that was so cold everything was freezing within minutes of going outside so everyone just let their horns freeze in a different position and we played the notes we could.