Again no idea about in the US, but in the UK at least, it's the marriage schedule that you and the witnesses sign, after which the certificate is produced. The contract is not a contract in the normal sense - it's not something that you sign, or get a copy of, it's more like a symbolic word.
None of this has anything to do with marriage. If you're still referring to the marriage contract, and the rights and responsibilities it provides, they aren't about obeying or controlling your partner, they're about things like financially supporting each other, child support for any children resulting from the marriage, rights regarding benefits and pensions that apply to married couples but not just co-habiting couples, etc.