Great question and actually a tincture is not the best way to preserve a chickweed tincture anyway, so great intuition there.
I would suggest making a vinegar, as you mention, or making ice cubes. Add chickweed to a blender, add a dash of water, blend this up and keep adding water until it is the consistency of a thin smoothie, pour this into ice cube trays and freeze. Once the cubes are frozen, drop them into a freezer bag, label and you have fresh chickweed juice to access throughout the winter. This could be used as a poultice or as a internal green juice ‘shot’. I also love doing this with cleavers and red clover, I tend to make a mix of the three.
If you want to have some for your skin, you could dry it out carefully and make an oil.
You certainly can dry some out as well to have to add to teas, but I find preserving fresh chickweed is great to have on hand in the winter.