Question:  If I sell a handgun via gunbroker to someone out of state does it need to be shipped to their FFL from a NJ FFL or can it be shipped by me to their FFL directly?

pistols need FFL. Rifles do not.
Or use shipmygun.com thru BUDS gun shop.

Legally, you can ship directly to the FFL.  It is up to the FFL whether or not they will accept the shipment from a non-licensee (best to call and confirm).  It is also possible that either UPS or FedEx (your only two choices for shipping it yourself) will give you a hassle about it.  They should not have an issue, as it is going to an FFL, but sometimes people get turned away by them.  It should go without saying, but you need to ship it via the actual hub - not a local store that's just flying either the FedEx or UPS banner.

Use Buds Ship my gun, they provide a letter from UPS that you can show to the local UPS depot if they give you a hard time about shipping, plus it's cheaper than doing it yourself