To clarify, a backdoor Roth conversion means making nondeductible contributions to the traditional IRA and then converting it to the Roth IRA. But since you have no earned income you cannot make any traditional IRA contributions.
Or are you saying you want to move more funds from the 403b to the Roth IRA? If you only have pre-tax funds in the 403b then the conversion to Roth IRA will always be taxable.
Any withheld federal taxes will be carried to line 25b of Form 1040 and applied towards any taxes due.
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