My tobacco stick fits better when I push it deep into the heating chamber. Will this damage my Ploom AURA?

No. This is normal and won't damage your Ploom AURA.

When you insert a EVO tobacco stick into your Ploom AURA, it gradually gets pushed deeper as you use it. This is totally fine and the stick being pushed in won't damage it or your Ploom AURA, unless you force it into the heating chamber.