OK, you have to consider the application of the rules to each child separately; they may not all qualify or they may qualify differently. If you answer the questions in the Turbotax interview carefully, you should get the correct outcome.
There are two types of dependent, qualifying child and qualifying relative. Qualifying child has the $2000 child tax credit, and also qualifies you for EIC if you are low income. Qualifying relative only has a $500 "other dependent" credit and does not make you eligible for EIC, or the dependent care credit.
For qualifying child, the child must be under age 19, not provide more than half their own support, and live in your home more than half the nights of the year (more than 183 nights). Because you are the grandparents, you have the same legal standing as the parents and you do not need to have the child via court placement; length of residency is the key factor.
The newborn does not have to live with you more than half the year, only more than half the days she was alive in 2018. If born in June and left in your custody in August, that seems to cover it, she is your qualifying child dependent. In Turbotax, select she lived with you "all year".
The oldest child who lived with you since May is also a qualifying child dependent if she lived in your home more than 183 nights. "About 6 months" is not specific enough. Count up the days. If more than 183, in Turbotax select "7 months" for how long she lived with you.
The middle child is more complicated. If this child lived with her parents more than 183 nights, then she is their dependent. You said the other grandparents had her 2 months and you have had her for 4-5 months, so it is possible that no relative had actual custody more than 183 nights, but you have to count the days specifically. If no one had custody more than 183 nights, then she is no one's "qualifying child" dependent. She can be your "qualifying relative" dependent if you paid more than half her support costs for the entire year. Support you provide includes the value of housing, medical expenses, travel, clothing and so on. (For housing, divide your total cost for housing, including food, utilities, mortgage, insurance, etc. by the number of people who lived in your home. If your total costs are $2000 and there were 5 people living in your home after the kids moved in, you would get credit for providing $400 per month of support for the months she was living there.). After you figure out the allocated housing costs, add any specific costs to this child such as medical care, travel, tuition, and so on. You will also need to know, or at least have a reasonable estimate, of the amount of support provided by the parents and the other grandparents.
If, considering the child's support costs for all of 2018, you can't prove you paid more than half, then you can't claim her as a qualifying relative dependent unless you have a signed multiple support agreement. This is a statement signed by all persons who paid more than 10% of her support, agreeing that you are the person who can claim her as a dependent for 2018.
The rules are here, and there is a worksheet to help you calculate the support question. https://www.irs.gov/forms-pubs/about-publication-501
One note about social security numbers. You can't claim any of the children unless you have their SSNs. If the parents will not give them to you, you may have to get the court involved. If the newborn does not have an SSN yet, you can't claim them until you receive it. You will need to apply for the SSN, possibly with the court's assistance. Wait until you have all the SSNs before filing, even if you have to get an extension and file late. If you were to file and claim the two older children because you are waiting for the newborn's SSN, and then file an amended return when the newborn's SSN arrives in the mail, the child tax credit for the additional dependent won't be paid. You must have all 3 SSNs before you file an "on time" tax return -- on time means before April 15, or before October 15 if you get the extension before April 15.
