I've been relocated for work that included long-distance moves in the past. Each time the total cost (paid by my employer) was in the 17K-20K range. That did include allowances for meals, lodging, gas, and temporary housing while homes were being built or ready, but those things combined were still a smaller amount compared to the packing, moving, storage, and delivery of my home contents.
I would not use an unknown, small company for a cross-country move. Too much potential for things to go wrong with too little potential for recourse if you need to recover costs for broken, damaged, or lost items.
Warning: If you buy insurance from the moving company, read the fine print carefully, ask questions and get responses in writing, and know exactly what this coverage actually means. You need separate, specific coverage for high-end items. You may also want to check with your homeowner's insurance if you have expensive items that the moving company will not cover or will not cover sufficiently to recover in the event of potential losses or more likely, damage. I've moved many times and have never had a move where something didn't get damaged, so just know that it happens.
We just moved last fall---three miles from our old house to our new house. It was $6000, so I think any quotes you get in the 9-10K range are reasonable for such a move.
Don't automatically go with the lowest cost. Instead, look for a company that offers the best services, and especially customer service, for the money you pay.
If you've never used movers before, be ready to be specific with them in your directions of what contents you want packed (vs. what you will be taking with you) and how the boxes are marked. Don't be afraid to step in and talk to them directly (in a polite way, of course) if you feel they are not treating your belongings or your home with proper care. Remember, you're paying them for that! Moving is difficult enough, and watching the entire contents of your life in the hands of strangers can be very stressful, especially for children. If you can have someone take the kids out of the house while they are packing up, it's much better. Then, you can focus totally on supervising the packing. Remember, it's always a nice to offer cold, bottled waters to the movers and buy their lunches---pizza or subs are usually much appreciated. That gesture goes a long way, I've found.
Good luck with your move!