Technically since you traded in a vehicle that already had an active policy, your new vehicle should be covered. Most insurance companies will automatically cover a new vehicle, for existing policy holders, for a few days until you are able to contact them to change vehicles on your policy.
There are a few things you will need before you take your new scooter out on the road to be legal in Delaware. You will need a motorcycle license and you must register and title your scooter with the state.
California does not allow credit to be used for car insurance purposes so you won't have to worry about that impacting your rate. Finding an affordable rate is all about shopping with as many companies as possible to find your best insurance option.
Your son would be covered as soon as he is added to the insurance policy so as long as the incident happens after the fact then he should be covered. That being said, if he files a claim within a short period of time after being added to the policy then your insurance company is likely to investigate further to make sure the damage didn't occur before he was listed as a driver.