Greetings @SingingWitch and congrats on your lovely new bracelets! 
My first impression was that they might be lava beads (as I have a bracelet in a similar style with lava beads), but on closer inspection, your black beads are smooth- so probably not lava stones! Apologies I’m also not a crystal expert, so hopefully those who are will have better guesses for you! 
As for cleansing, many witches do like to cleanse under the light of the Full Moon. However, you could try cleansing/charging your tools with Sun Magick on any sunny day 
There are also plenty of ways to cleanse tools that use forces other than moon magick- you could cleanse with salt, do a sage cleanse, or use palo santo.
I hope you can figure out the identity of your lovely bracelets and that you find a way to cleanse them that works for you! Good luck and blessed be 