I recently completed ODS in August. I felt a little confused about the whole process, so I'd like to clear up some uncertainties that I had felt before, just like you are feeling now.
I graduated in May but went to the August ODS. The timing is all about when you pass your NCLEX and when you get commissioned. If you pass your NCLEX and get commissioned early in the month, look towards heading to the ODS the following month. (I graduated in May, passed the NCLEX in June, got commissioned in July, and went to ODS in August)
As for the HHG: you mostly have 2 options as nursespouse has stated. You will get BAH. I find it rare for officers to live on base. I did the DITY move and was reimbursed for my expenses. Keep in mind that they will not ship your personal vehicle, so you will need to drive it to your first duty station.
Another option is to keep your car and drive to ODS in Newport. Then they will give you multiple travel days after you graduate from ODS to travel to your first duty station (depending on the distance from your duty station from ODS) If you travel by air, you have 1 travel day.
You will find out the huge difference between ODS and OCS.
1. You are a commissioned officer when you arrive in Newport, thus you are being "developed"
2. Officer Candidates are seeking a commission, so their process is mentally and physically tough to find those most suitable to become an officer.
If you have any other questions, feel free to contact me.