The safety issues faced by people displaced by disaster are multifaceted. Even if you seek shelter in an 'approved' Red Cross or other organizational sponsored shelter, you need to remember - "Safe" is relative term.
For this reason, I'll begin by saying, if you don't have to leave your home, don't.
Weather extremes - hot or cold, are the primary reason I suggest knowing where your nearest shelter is located. These normally have at least minimal facilities for heating/cooling and basic sanitation - normally.
Once you leave your home, you have become a "refugee". So, what to do now?
Once you have found shelter, you need to decide how you will ensure your own safety. Few public "shelters of last resort" will have an assigned security staff or police - and are intended for very short term use. This means you are on your own. Determine where the exits are located and if they are actually operational. If you are traveling/sheltering as a family, plan on one adult staying awake as the others sleep.
Rarely do public shelters have cots, bedding or blankets. More organized areas and Red Cross shelters do have cots and may even have blankets. This is why you have a blanket in your DIY kit.
Sanitation will be a big issue, so bring your own toilet paper and hand sanitizer. Putting these in a small bag or purse makes it easier to carry. Understand now, that at unstaffed shelters, the sanitation faculties will get nasty - fast. Women may want to consider carrying one of any number of available 'sanitation devices' or female urination device (FUD) (this is how they are found on the Internet) which will allow use of the facility without the need to make personal contact with a filthy toilet surface.
Water from public sources (water mains) may be compromised after a disaster - unless you know the water has been treated, any water obtained from drinking fountains, facets or hose bibs must be treated. The water tote in your DIY disaster kit will make a great 'holding tank' for the time it takes for your treatment method to take effect.
Public shelters may or may not serve cold meals.
A shelter will be noisy. A few sets of foam earplugs will go a long to allow you to sleep when it's your turn. For the same reason, a set of earbuds for your radio will go a long way to reduce tensions in a shelter area.
Finally, most shelters are in public schools, so find a comfortable corner, set up in the corner and be prepared to make the best of it.
Your valuables -
You may not have the paperwork in hand, but a digital copy is normally enough to at least get started on the claims process. The price and size of these storage devices have fallen, so more than one copy is not only possible, but recommended - but take the time to put some kind of simple encryption on the device is case you lose or it it is stolen. The internet has many, many sites that describe just how to encrypt this kind of data. Find one that matches your operating system and hardware. One last thought - save your 'paperwork' as an image file onto a SD chip device. If needed, a one hour photo can 'print' out your paperwork for claims...
Carrying large amounts of currency after a disaster is almost a necessity - if the electricity is out, credit cards are worthless.
When I say security, most folks I know think of some kind of firearm. And while being armed can provide a sense of security, please take some time to think things through. Once you pull the trigger, you own that bullet until it stops. And after. Don't believe that a shooting event in a disaster will be treated differently - it will not. In the afterwards, and there will always be an afterwards, you will have to face the results of your actions.
Safety once you return home.
Home security. This is another area where folks might think a shotgun is all that is needed. I would suggest a bright search light/floodlight and someone to use it will provide a lot more security for your belongings. Thieves will generally stay away from an area they know is being watched. So, again, one adult should plan of being awake all the time until things return to 'normal'.
I hope this segment has given you some things to think about now - and the push to add these to your overall planning.