I know something is wrong because my tax due increases, that shouldn't happen in backdoor conversions (when triggering form 8606). Thing is, I can only trigger it for 2024. How to do it for 2023? (without any tax changes in my 2024 declaration)
Roth contributions spread across 2 years are a 2-step process. For the 2023 tax return, you should have prepared a Form 8606 - check your tax return to verify you did. If not, you can do so now. There may be a $50 late fee assessed unless you have cause. See: About Form 8606, Nondeductible IRAs to get a pdf form and fill it out, if needed. The 2024 8606 will then have the 2023 contribution reported on line 2 of the 8606.
Next, since the conversion for both years happened in 2024, it is correct that they were both reported to you on the 2024 1099-R. In order to ensure no tax is computed on the conversion, since you did not deduct them in the first place, after you enter the 1099-R, continue through the interview until you are asked f you moved the money through a rollover or conversion. Answer that you converted (all of it) to a Roth IRA.