You should be able to get your own plan. Because you are in New Jersey, all people in the same household must be on the policy unless you can show proof the other people you live with are insured elsewhere.
I would call up your local DMV or appropriate state department and ask them what to do. Most states allow you to pay a small fee and show an out of state status and they will issue you a temporary paper plate for the vehicle to be relocated.
Insurance companies (depending on your policy and whether you have replacement cost value coverage) are obligated to bring your property back to its previous state, but no better. This, in theory, means that it should be no worse as well.
Switching insurance companies won't necessarily raise your rates as long as it doesn't result in a lapse in coverage. Accidents and violations can stay on your record for up to five years, so your past infractions from six years ago should not affect your rates.
You will need to get a commercial policy since you will be a delivery driver. If you were going to drive for a ride share company you could get a personal auto policy with an endorsement to drive for ride-share.
Since she lives with you and will be driving your car, you will want to add her as a driver to your current policy. This is typically the most cost-effective as well since you already have a policy established.