If you have no traditional IRA funds, then you can do a "backdoor Roth conversion" fairly easily. You make a non-deductible contribution to a traditional IRA, then after the transaction has settled, you roll it over to a Roth IRA. Any gains are taxable, but the contribution balance is not taxable. You can do this any time from days to years later. But most people do it within a few days, so there aren't any earnings to worry about. (You can specify that the money goes into a cash account so it will only earn a very little or no interest.)
You can't do the conversion for 2023, that only happens on the date it happens. It's not retroactive like contributions. In your case, you could contribute up to $6500 (or $7500 if over age 50) as a retroactive 2023 contribution (before April 15), then do a rollover to Roth. Then later in 2024, you can contribute up to $7000 (or $8000) for 2024, and roll it over. Both conversions will be reported in 2024, since they are reported when they happen, even though one of the contributions will be reported in 2023. Your spouse can do the same, except part of their conversion will be taxable.
For your spouse, they will have to convert their traditional IRA funds to a Roth IRA and pay income tax on the conversion, either before or as part of the backdoor conversion. (This means all traditional IRA funds. If you have IRA accounts at more than one broker, they are all added together for determining the tax consequences.) If you can't pay the tax all at once, you can do it over time, but it will be pro-rated. For example, suppose your spouse has $10,000 in a traditional IRA. If you contribute $6500 of new funds toward 2023, and then convert it all in 2024, $10,000 of the conversion (about 60.6%) will be taxable, and you will likely owe $3000+ in state and federal taxes on your 2024 return (depending on your tax bracket and other factors, of course). From then on, the backdoor Roth would work like yours. If you only converted the $6500, then $3900 (60%) would still be taxable income, because conversions are pro-rated, and you would be considered to be converting $3900 of the original pre-tax funds and $2600 of the non-deductible funds. After the conversion, the remaining traditional IRA balance of $10,000 would contain $6100 of pre-tax funds and $3900 of non-deductible funds (because a partial conversion is pro-rated). This would go on until you convert all of the traditional IRA to a Roth which will eventually zero out the pre-tax balance.
