Честита Баба Марта, Каролинке!
There's no particular "etiquette" with martenitsi. And it's not realy a gift, you're not giving people a martenitsa to have. It's a longevity and happiness charm, a ward against bad stuff. So it doesn't matter if it's an old martenitsa (though you should've put those on a blossoming tree or under a rock). What's important is not the physical thing, but the effect and the gesture. That's why you should tie or pin them on people and not just hand it over. Though I imagine with your students that could take a while :)
On March 1st and the next several days I walk around with some extra martenitsi in my pocket, in case I meet someone I wasn't expecting to. There's no reason not to tie a martenitsa on someone you know. But you're not obliged to, because it's pretty certain they already have several. The idea is that by the end of the day everyone and their dog (literally) should have a piece of red and white string for health, long life, happiness, unity, good fortune and nice weather in March.

I remember last year when you wrote there are Chinese martenitsi being sold. I was surprised and this year apparently it's worse. From what I've seen, there are four types of martenitsi - proper martenitsi, incl. Pijo and Penda (not Piju and Panda :Р), weird martenitsi, martenitsi that are looking at weird in the rear-view mirror and martenitsi that are starting to be misguided.

I like your rule about bying - I think I follow it too.