You do not need dependents for any of the stimulus payments. At this point if you are otherwise eligible, you can file a 2020 tax return to claim the Recovery Rebate Credit for stimulus rounds one/two (1200/600). You will file a 2021 tax return to claim the Recovery Rebate Credit for stimulus round 3 (1400). Here is the IRS FAQ for reference.
Generally, if you are a US citizen with a valid social security number and made less than $75,000 and were not a dependent on some else's return, you are eligible for the Recovery Rebate Credit. Here are the eligibility requirements. See questions B1 and B10.
TurboTax will ask you about any stimulus payments you have received in the review process before filing your returns. On the 2020 return you will be asked about rounds one/two and on the 2021 return- round three.
You can start your 2021 return with TurboTax here and 2020 here.