Your husband may be trying to discourage you from hyphenating your name, but under the law there is no penalty at all for doing so. As for what happens to your husband's assets in the event of his death, your name has nothing to do with it. What will happen will depend on whether your husband has a valid Will and/or other valid estate planning documents in place (such as revocable or irrevocable trusts), what those documents say, how his assets are owned, and how any beneficiary designations are set up.
Assuming that your husband has his principal residence in Georgia at the time of his death, if you and his children all survive him, then you and the children would all be his heirs. If assets become part of his "probate estate," then if he has no valid Will, you and the children will split the probate assets (after any debts, administrative expenses, and taxes due at his death are paid), except to the extent you (or any minor child) is able to make a successful claim for a year's support. If there are still only 3 children, you would get a third (plus any year's support award, if you get one -note: the year's support is not a set amount and not guaranteed if the children challenge your request), and the children would divid the other 2/3 between them equally. If your husband has a valid Will, then his probate estate will be distributed in whatever manner the Will says it gets distributed. He could try to disinherit you completely, if he wanted to.
Assets which do NOT become part of the probate estate would include: (1) any assets your husband owns jointly with you or anyone else as "joint tenants" - usually, a joint bank or brokerage is owned as joint tenants but real estate is only owned that way if the deed SAYS "joint tenants," "with rights of survivorship," or something very similar on it; and (2) any assets which pass to you or someone else at his death under a beneficiary designation - usually, life insurance, IRAs, 401(k) accounts, and other retirement accounts have beneficiary designations - or a "payable on death" or "transfer on death" designation. Note, however, if your husband's estate IS the designated beneficiary of an asset, then that asset WILL become part of his probate estate. But in general, a right of survivorship will result in that asset passing outright to the surviving joint owner at the first owner's death, so that asset would not become part of the first owner's probate estate. Similarly, if an individual or a trust is designated as the beneficiary under any type of beneficiary designation, that asset does not become part of the owner's probate estate. If your husband WANTS to have his assets divided among you and his children, he can, and if he does nothing, his probate estate will be divided that way. But that has nothing to do with what your name is, except to the extent, if any, that he decides to use a different estate plan if you hyphenate than he would if you didn't hyphenate....
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