Your child must check the box that says "I can be claimed as a dependent" if they can be claimed, even if they don't want to be claimed. Unfortunately, unemployment compensation is "unearned income" and dependents don't get the usual standard deduction applied to unearned income. However, $6000 in tax seems impossible, that would imply total income close over $40,000. Even if it was $3000 per child for two children, that would imply income of $25,000 per child, counting wages and unemployment.
Your child who lives with you can be claimed as a dependent if they meet the following tests,
1. Qualifying child rules
- Age under 19, or a fill time student under age 24
- Does not provide more than half their own support
2. Qualifying relative rules
- Any age
- You provided more than half their total support
- Their taxable income is less than $4300.
(There are other tests that I did not mention because they already meet them as you described your question.)
You will need to carefully consider their income, and how much support you provide them (including the value of room and board) compared to how much of their own expenses they pay. Given the amount of income they seem to have, I think it would be hard for you to be providing more than half their support, unless you live in a mansion where your mortgage is so substantial that 1/3 or 1/4 of the mortgage payment allocated to each child is more than their total income.
