Can anyone assist me with submitting a CWO package for the Army, as a service member that serves currently in another branch?

Into what specifically? Each WO MOS is different, you need to be addressing this to a specific field. I.e. Aviation, you need to try contacting the WO Academy. keep in mind each branch proponency sets the requirements for their related career field. Hence no two warrant schools have the same requirements.

https://usacac.army.mil/organizations/cace/wocc

You can contact warrant recruiters to get all the latest and greatest information and assistance with this. They are usually colocated with SF recruiters. Depending on where you are currently stationed, you may be able to walk right in and get the process started. I will tell you right now, from my own experience, you better be ready for a long process; just getting the command recommendations and letters of recommendation from the WO in your chosen field is a pain in the ass. I was pretty close to finishing up my packet when I was recommended for medical retirement.