I assume you have restored your license. NJ bills you $1000/year for three years. That needs to be paid. It looks like only 2 may have been billed. (2011-12) Go to MVC and see if you are in good standing. If you are the transfer is no problem. NJ would count the DWI as a first if you were ticketed for a second. I do not know if Ca. does the same.
When you say that all is paid and clean, I am assuming that you mean you have paid all fines and have served your term of suspension, have paid your MVC surcharges, and done your IDRC. If you have done these things AND have restored your drivers license, then you should have no difficulty getting a California license when you get there. However, if you are still suspended, then you will have difficulty getting a license. Of course when you get to California, you will have to take a written examination to get a DL.
I can only tell you what will happen in New Jersey. I cannot tell you what California will do as I do not practice there. California is a signatory to the interstate driver compact and will be notified of your record. I would recommend that you contact a lawyer licensed in California to get any of the local procedural wrinkles. I would also contact the NJMVC and confirm that you are restored and in full compliance. You did not say whether you have a post suspension interlock requirement. That could complicate your licensing.
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Each state enforces their laws differently, including DWI/DUI. How a state treats an offense from another state also varies. For example, New Jersey has reciprocity with most states. Therefore, if you are suspended in New York for a DWI, New Jersey will also suspend your ability to drive. The motor vehicle commission or governing agency can inform you as to the treatment of this issue, including how a driver's abstract will reflect the suspension.