Do I have to submit a packet to be released from my unit to go active navy? Or is the 368 all I need?

The USAR requires a big old packet to be released right now. There's no guarantee you will be released and the USAR is under no obligation to release you.

From the USAR Personnel Action Page - 1. Soldier submits DD Form 368 completed thru Item 4 and completed DA Form 4187 with justification. Unit submits thru Chain of CMD to GOCOM CDR (MSC) for final approval. 2. If approved, the DD Form 368 is returned to Soldier to take to recruiter.
[NOTE: Then follow with remainder of existing instructions.] 3. Approved conditional release along with supporting enlistment documentation is forwarded to the RPSC for publication of appropriate transfer or discharge orders (and certificate). The RPSC uploads the discharge or reassignment order and the DD Form 368 are uploaded to iPERMS. 4. The RPSC makes distribution to Unit, Soldier, GOCOM iPERMS and the USAR Pay Center. 5. The RPSC updates the personnel database.
(Documents include the 638 and 4187 in addition to DA4856 with command and retention and a justification memo/letter from you.)

SPC (Join to see) Below is one post dealing with your question.
https://www.rallypoint.com/answers/army-reserve-looking-to-go-navy-active-duty

The DD368 + what ever else your unit requires for the approval authority to decide= packet.

Sure do. The ARMY recruited and trained you, the reserve does not have to release you from your contract just so you can go to a competing branch. You have a better chance of going Active Army than going Active Navy.