A TD visa (I looked this up the other day) is a visa issued to dependents of Canadians and Mexicans who have a TN visa issued under NAFTA.

You should be able to get TINs for them, as that is a tax id number, and I don't think you even have to come to the US to get one. You need one to open a brokerage account, for instance, and a lot of foreigners have bank or brokerage accounts in the US. The TIN is used to make sure they pay their taxes on monies earned in the US. They are issued by the IRS, not Social Security and the one you would want is the ITIN. See here.

Another option is to get an expat insurance policy to cover your family until you are eligible for the Hawaii insurance.

I assume your employer doesn't allow family coverage, or that is what you would be doing. If you are self-employed, see if you can find an association to join that will offer coverage.