How do I go about switching services (active duty Army to active duty Air Force)?

You have to pass the test...

You need to work with the Air Force recruiter... The Army will give you the run around...

Almost certainly start by contacting an AF recruiter. Good luck!

There was a DoD policy that authorized the "Blue to Green" program that allowed AD SM's in the AF and Navy, who were identified for RIF because of their MOS, to transfer to AD Army for the remainder of their contracts. There is no such program or policy for AD Army to transfer to another service. You will have to fulfill the remainder of current Army contract, then apply to enlist in the AF as a prior service applicant. All services have there own policies on enlisting prior service, and there is no guarantee that the AF or any other branch will be accepting prior service at the time you apply.

The Army is in retention mode right now. They are paying Soldiers to stay in uniform again. Yes, the USAF is desperate and has rescinded their tattoo policy and eased their rules on previous use of ADHD medication. However, they (the USAF) will not talk to you until you are 90 days out from your ETS.

Is a branch transfer possible? Yes. That's what the DD Form 368 is for (Conditional Release). Have I ever heard of it actually working? Nope.