Keep a level head:
1. Get/renew your passport. Unlikely to be needed for anything other than leisure, but prudent to have.
2. Talk with a financial advisor about diversifying your portfolio. Make sure your financial investments aren't completely US dollar based. In other words, make sure you have investments in foreign markets. Realize the bulk of your investments will still be in American markets.
3. If you live in an area that has required FEMA assistance, realize that assistance may not be forthcoming. It's never a bad idea having a few days of food and water on hand in case of emergency - but you're preparing for natural disasters, not political disasters. Just expect to deal with natural disasters on your own with little to no federal assistance.
4. It never hurts to invest in defense. You're not preparing for war, but since you may be on your own when dealing with natural disasters that may result in increased looting, you need to be able to defend yourself. Whatever defense you invest in, make sure you've also invested in the know-how for how to use it.
5. It never hurts to keep more cash on hand, as you've said. Consider keeping a small cache of cash safely in your home. If you need to make a quick getaway due to a natural disaster, it's advantageous to have some cash. As you've mentioned, keep more cash in your bank as well to help weather larger storms.
6. Grow your own food. Develop your green thumb. Gardens kept a lot of people fed during the Depression. Learn how to can as well.
7. Work on having a survival pack. Collect everything you need for an emergency and have it ready to go. There are a lot of "prepper" YouTube sites to give you ideas of what's useful to have in the event of an emergency.
8. Identify another area in the US you could go where you would be impacted less by these kinds of events. Ideally, you should arrange to go there before you need to, but at least have it in your back pocket.
9. Doesn't hurt to identify a country you could go to should it be advantageous to leave the US for any reason. Keep in mind that all Americans are thinking Canada and Western Europe. I would recommend finding a place that would be thrilled to have you and would be more advantageous to be there than the US. Make sure if you have to leave for whatever reason, you won't be jumping from the frying pan into the fire. While it may be rare for most people, you may find it advantageous to emigrate there now.
10. Don't lose hope. Though the Depression was certainly bad, there was still business being done and gains to be had. Take stock in the things you enjoy and ensure you have them in your emergency pack.