If your unmarried partner had income from working, then there might be benefit to the partner claiming some of the children as dependents on their tax return while you claim the other children on your tax return. There are a lot of factors and it is really impossible to figure out without actually testing different combinations.
But if your partner does not have income earned from working, then the only filing solution that makes sense is for you to claim all the children as dependents, and you probably also qualify to file as head of household. If you also claim your partner as a dependent, you would be eligible for up to an additional $500 tax credit for "other dependents."
However, the credit for other dependents is non-refundable, that means it won't be paid or credited to your account unless you owe at least $500 of income tax. Since you have 3 children, who will qualify for up to $2000 credit each, and your total income is only $30K, it seems most likely that claiming your partner as a tax dependent will not actually change your tax or refund. You can try in the program, but it may make no difference.
We can't tell you if other government benefits might be at risk if you claim your partner as a dependent. We don't know what benefits you are talking about or what the rules are. If these benefits depend on the claim that no one else pays your partner's expenses, and that is untrue, then the benefits could be at risk no matter what you put on the tax return. And it is also true that if you file a state tax return claiming your partner as a dependent, that information could be shared with other state agencies.
