Q. Am I within my rights to claim them as dependents given the above info?
A. No.
You cannot claim your GF's children because they are not related and did not live with you all year. As others have described there are times when you can claim them, but not as qualifying children for the EIC and Child tax credit . Even in the rare case when you can claim them, you can only claim the (non refundable) $500 other dependent credit.
Since you are splitiing up, you cannot even claim your own child, since he/she will be living with the mother more time than he lives with you (she's the custodial parent). She can release the dependent (and the child tax credit, but not the EIC) to you by signing and giving you IRS form 8332. https://www.irs.gov/pub/irs-pdf/f8332.pdf
*There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.
So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
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