I'm Lucy, and I'd be happy to answer your questions today. Congratulations!
If you BOTH noted on your marriage license that you intended to use the shortened name going forward, you can provide take copies of the marriage certificate and your identification down to Social Security to get new ID cards issued, and you'll be able to change the name on your driver's license/other documentation just with the marriage documents. Check with each agency before contacting them to see what documentation they need - some want an original marriage license, and some keep it, so you may need more than one.
Name changes made due to marriage that are not noted on the marriage certificate must be made by going to the local family court and filing a Petition for Change of Name. In that scenario, you wouldn't be able to get new ID in your married name without a court order, even though the change was due to the marriage.
There are no laws regarding what a parent can name a child. The child does not need to share a last name with either child. You're allowed to write whatever you like on the birth certificate.
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