Retirement tax questions

As I see it, New Jersey has assessed the all contributions for NJ Income tax at the time of earning, irrespective of the fact that the same contributions are at the Federal level deferred compensation and not taxed until withdrawn from the savings plan.

Massachusetts, instead, defers assessment on both the employee and employer contributions until such time as withdrawn.

"Distributions made to you from the 403(b) plan are excluded from gross income if such distributions represent Massachusetts previously taxed contributions.  Distributions in excess of Massachusetts previously taxed voluntary contributions before January 1, 1998 are taxable"
https://www.mass.gov/service-details/view-government-pensions#403

Apparently you retired or otherwise ceased to continue participation by contributing after relocating to Massachusetts.  Just in case that is incorrect and you did make contributions when a Mass. resident  that distributions made to you from the 403(b) plan are excluded from Mass. gross income if such distributions represent Massachusetts previously taxed contributions.

Several key points:

1. A resident of NJ having relocated from Massachusetts and then receiving pension distributions does not  have a total reciprocity discussed in this article following.  Massachusetts allows a deduction for some or all of the distribution depending on the other state's (NJ) treatment of MA pensions:
   New Jersey All out-of-state government pensions qualify for the pension exemption:
   For the 2018 tax year in New Jersey, if you are 62 or older and make less than $100,000 per year, you can exclude up to $45,000 of pension, IRA or annuity income if you are a single person. Married and civil union couples filing jointly can exclude $60,000, and those filing separately can exclude $30,000.
   https://www.mass.gov/service-details/view-other-states-tax-treatment-of-out-of-state-government-pens...
   https://www.state.nj.us/treasury/taxation/njit7.shtml
   
   
2. Enacted in 1996, the Pension Source Act (P.L. 104-94) prohibits the states from taxing pension income of non-residents, even if that pension was earned in the non-resident state,
   
   
3. Massachusetts generally follows the rule for calculation distributions from a qualified retirement plan in which you have "basis" - that is already taxed monies. The distributions become taxable after exhausting the basis - such that it is only the generated income within the plan that is then taxable, but there must be a valid accounting.  So, that said, the question to which there seems to be neither a MA DOR TIR or other statement is:
   Since all the basis was of non-Massachusetts earning income, should that basis be treated in a similar fashion?
   I would argue yes and that would mean that in the MA Form 1 or Form 1-PY/NR interview you would indicate that basis in the calculation of what is taxable in the distribution.

From what you wrote, this is probably redundant, but note that a non-resident of Massachusetts is not subject to tax on pension distribution https://www.mass.gov/info-details/learn-about-tax-treatment-of-pensions-in-massachusetts#nonresident...  as this reiterates the (P.L. 104-94).