There are no timing restrictions on when you can perform Roth conversions. You can also roll money from a 401(k) plan or the federal TSP directly to a Roth IRA in a taxable rollover at any time that you are permitted to distribute this money from the 401(k) or TSP. To make this be income in your new state rather than in your old state, these Roth conversions or rollovers to the Roth IRA must occur after you establish residency in the new state.
Not sure what you are referring to as a backdoor conversion. You haven't mentioned making any nondeductible traditional IRA contributions.
