Assuming you are eligible for premium tax credits, the amount of your credit will be calculated based on how you file your taxes. If, for example, you each claim one of your children, you each will be considered as a household of two. The income of each household would be evaluated separately to calculate eligibility for and the amount of premium tax credits and cost-sharing reductions. Using a different example, if you claim both children as dependents on your tax return, then you and your children will be considered a household of three, and your income will be the basis for determining subsidy eligibility for the three of you. Your partner will be a household of one and his/her eligibility for premium tax credits will be determined separately.
As for the type of coverage your family can purchase, that may vary based on the marketplace rules where you live. Currently, the IT functionality available through the federally facilitated marketplaces cannot support people in different tax households enrolling in the same policy together. Information may still be collected about other members of the household, even if they are in different tax families (for example, unmarried parents who live together but file separately). The marketplace will ask for their relationship to members of the other tax family and their income since it may factor into Medicaid eligibility determinations. The other household members will be included in the application as non-applicants. (Center on Budget and Policy Priorities, The Health Care Assister's Guide to Tax Rules, updated Sept. 2018; CMS, Application Spotlight: Family and Household Composition Section, Oct. 2017).