Worried about crossing on my own with H4

My children and I (all H4 visa holders) will be crossing into the states without my husband (their father and H1B visa holder). Is CBP going to have an issue with me crossing without him? I do have a travel letter coming from our lawyer and signed by husband stating I'm travelling with them and he knows.

The other part of the issue that is complicated is we are going to AZ for Christmas and the kids and I will stay there until we find a home in TX. Will they ask a lot of questions and have a problem with us going to AZ when the H1B job is in TX? I'm extremely nervous as this will be my first time crossing since we got the H4. Just wondering what sort of questions I should expect?

No worries. The letter covers travelling with the kids internatioanlly.

You got H4s for a reason: to enter US without problems.