Maybe. For 2021, you are still Nonresident Aliens for tax-filing purposes. Although each of you received OPT status, this is still considered an "exempt" status for residency purposes. Since both you and your fiancée did not receive H1B status until September and did not have any nonexempt residency period in either 2020 or 2019, you will not meet the substantial presence test of 183 days in 2021. You will under normal circumstances both need to file Form 1040NR this year.
However, if you do anticipate that in 2022 you will meet the substantial presence test, you have the option to elect dual residency status. Please see example #8 at the following website, which best applies to your situation:
Tax Residency Status Examples | Internal Revenue Service
If you elect to file as a dual-resident alien, then for the portion of the year in which you (or your fiancée) earned US-sourced income while on F1 nonexempt status will be included on Form 1040NR (not supported by TurboTax), but the remaining income after your elected residency start date would be filed on the US Resident Form 1040. When you file as a dual resident, the 1040NR is included as an attachment with Form 1040. For assistance in preparing Form 1040 NR, see this Help Article: Does TurboTax handle Form 1040-NR for nonresident aliens? If you or your fiancée make the election for dual-resident status, you may use TurboTax to prepare the "resident" portion of the return. You will select the option to print and mail in the return, you will not be able to e-file your 2021 tax return. In 2022, presuming you meet the substantial presence test, you will file Form 1040 as US residents.
For your state situation, you are part-year residents of New Jersey and Washington. If you had no work while living in New Jersey, you won't have a state return to file, because Washington does not have a state tax return. If you worked while in New Jersey, you will file a New Jersey part-year return to claim the income you earned while in New Jersey. Your Washington income is not taxed in New Jersey.
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