I will try to include something on breathing in subsequent videos. Thank you for bringing this up. Meanwhile try to breathe through your nose while doing your adavus. It is a difficult transition to make sometimes, but it is important for aesthetic reasons. If you keep your mouth open to breath when dancing, it looks strange. Part of our job is to make it look effortless even when we are dying! :-)
Now to address your issue of going out of breath. This will continue to happen as your muscles adapt and become stronger. It will take a while for this to change. Once your muscles become stronger and your stamina builds, your breathing will become easier.
We tend to purse our lips when our physical effort is high. Once your body has adapted to a demand, it wont be as difficult physically and then your lips will relax.
I hope this makes sense .
Glad you are practising and thinking about these things