Can an old rosette “iron” that's starting to stick be reconditioned?

I am going to assume that your chrusciki is cast iron. I have been collecting and using vintage cast iron for 30 plus years and use the following method for reconditioning pieces which have lost their seasoned finish. Your irons must be able to separate from their handles, unless the entire piece is cast iron.

1. Separate all non-cast iron pieces from the cast iron pieces.
2. If there are crusty bits stuck to the cast iron remove these. I use a variety of methods including fire, scraping, and soaking in a hot water bath for 15 minutes or so. Soapy water is only okay before skillet is seasoned, never after. Keep the cast iron off porous surfaces during the soak as rust may form and be difficult to remove from them
3. Dry cast iron pieces and apply heavy coat of fresh cooking oil to entire piece.
4. Bake on a foil covered cookie sheet in a hot oven (350-375) for up to 1 hour.
5. Let cool, pieces may be sticky and this is normal.
6. Store on wax paper until first use.
7. Use as recipe suggests. Clean up irons using hot water only and a cloth of some type.
8. To keep seasoned, dry after cleaning, wipe with a paper towel dipped into fresh cooking oil, store away from moisture. Pieces don't need to be dripping with oil, just lightly covered completely. Can repeat steps 3-6 once yearly.

Cast aluminum molds for this sort of cookie do not season in the same way as cast iron. It doesn't rely entirely on polymerization of oil upon the metal for a nice release, it simply needs a very clean and smooth surface for the cooking oil to cling to before adding the batter. Over the years, small burnt-on bits and oxidation may accumulate upon the metal, making a clean release hard to achieve.

It sounds like it simply needs a good and thorough cleaning with a long soak in hot vinegar. If that doesn't work, a kitchen rag doused with lemon juice and sprinkled with table salt is the next step, be gentle and get inside all of the nooks and crannies. If that fails, A metal scouring pad, or even a wire kitchen brush, may need to be employed, but take care not to leave deep scratches. The idea is to smooth the surface.

Don't clean it with soap after using it again, which may damage the aluminum, instead wipe away any cruft with a paper towel and leave it to drain on a stack of paper towels. The oil will protect it from tarnish. (We'll just call it tarnish. Aluminum and oxidation in cookware is too complex a topic to cover in this answer.)