I'm Lucy, and I'd be happy to answer your questions today. I'm very sorry to hear that this happened.
If you did not purchase additional insurance, then you're unfortunately not covered for more than the basic insurance provided for in your contract. A judge cannot order them to pay more than the amount agreed to in the contract. On top of that, when a person goes to court, they're not entitled to collect sentimental value on lost or destroyed items. A judge can only award you what those items would have received if you were selling them at a yard sale or on eBay. It's extremely difficult to put a value on things like pictures, and the amount you'd get would probably be only a few dollars. For household items like dishes, clothes, and things like that, you're entitled to the depreciated value of the used items. That is NEVER the same as replacement cost, unless you have receipts proving that you bought something recently.
The first step is to contact the insurance company to find out how much they are offering to replace your damaged items, and consider it in light of the above. Ten boxes worth of items of primarily sentimental value are not likely to get you more than the Small Claims Court limit if you have to sue - that's true in both Texas and California. Unfortunately, that also means that it's quite possible the cost of hiring a lawyer to represent you would exceed the amount of money the judge would award you. The moving company does not have to pay your legal fees if you win unless the contract says they do.
Also, check the contract you signed. You may be required to file a lawsuit in California, and you're not allowed to collect the costs of traveling to California to sue. California also doesn't allow lawyers in Small Claims Court, so you'd have to represent yourself. That's important information to have before you start negotiating with the moving company, because you want to be able to weigh the expenses associated with collecting and the amount you're likely to get against the amount you're reasonably likely to recover.
If you have any questions or concerns about my response, please reply WITHOUT RATING. It's important that you are 100% satisfied with my courtesy and professionalism. Otherwise, please rate my service positively so I am paid for the time I spend answering questions. If you are on a mobile device, you may need to scroll to the right. There is no charge for follow-up questions. Thank you.