Yes, you may carry on your nebulizer. As a reminder, within the US a carry-on filled with medical equipment & so labeled does not count as your allowable carry-on, nor is it subject to a baggage fee. It still must meet the requirement of fitting in the overhead compartment. Any medications may be placed in there as well, but leave your saline, albuterol, etc in the original labeled boxes or pouches and allow extra time for scrutiny at the TSA checkpoint - it may not happen - but on my last 2 flights, that bag was thoroughly torn apart & I had to explain the saline vials as they were just in a baggie.
Happy travels.