There is no double taxation. The $6,000 of basis in nondeductible traditional IRA contributions that transfers from line 14 of your 2021 Form 8606 to line 2 of your 2022 Form 1099-R will reduce the taxable amount of the Roth conversion performed in 2022.
A Roth conversion has no bearing on your eligibility to make a nondeductible traditional IRA contribution.
A backdoor Roth is not a Roth IRA contribution. It's a traditional IRA contribution followed by a Roth conversion. Each of these transactions is reported separately.
