Sharps containers are just about everywhere if you look. Hotels, airports, stores, etc. Usually in public rest rooms. I'd skip carrying a sharps and just start looking for sharps locations. If you can't find one, ask someone who works there. Worst case, you repack some used needles for disposal when you find a sharps container.
I don't have experience with having to use sharps containers but my wife has an injectable migraine medication and I have to inject her.