Writing post-apocalypse/dystopian future can actually open a large range of possibilities and original ideas but there are some things we should try to keep in mind. The problem with this genre is that is a heavily research-based genre, it requires a lot of information so a lot of things can slip up or be forgotten.
Limits of the Body - This is something that seems to be largely forgotten unless in extreme situations. Yes, humans are tough, but they are also weak. We have that balance of nature within us. People can keep going whilst in a lot of pain, but something like a headache could knock them senseless. I knew someone who got shot in the leg and ran three miles because he thought he had been hit by the brick. The second he realised he had actually been shot and was safe to do so, he collapsed in pain. Quickly establish what your characters can and cannot do.
Children - Children are more robust than so many people give them credit for, they wouldn’t make it to adulthood otherwise. Children are emotionally stronger than they are physically but many children have a lot more endurance for their size than adults because they have to keep up with adults. Two good examples are The Road by Cormac McCarthy and Lord of the Flies by William Golding.
Hygiene - Namely with periods and stuff. It is hard work keeping the body clean, so personal hygiene will be poor but people tend to stop caring at a point when they realise how hard it is to maintain. A lot of people would revert to old fashioned methods of vagina health as well, so people would use reusable cloth or diva cups. The only book I know of that covers this is The 5th Wave by Rick Yancey.
Simple Killers - More people died of the Spanish flu than in the First World War (source). It is surprising how often it is “little things” that kill people off. An emotional death does not have to be all that dramatic, no bloody death or major killer. Something as simple as a small cut that festered will kill someone if not treated correctly. An asthma attack, diabetes, things we see as treatable would make quick work of us without medical aid. Also most deaths are really simple and sudden.
World Limitations - What is this obsession with guns and everything happening in the summer and unlimited cans of food? No, let’s be honest, none of these are realistic. Guns will not last, can goods will be snatched up by the shop loads, most natural disasters will happen in spring or autumn. Remember to do the key thing, make your world real and people are more likely to believe it.
Take a Page - Who are most renowned for their post-apocalyptic stories? British pre-1950s authors. Why? Their worlds are real, the possibilities of what could happen in those worlds are real. Some where and still are scarily accurate, they looked at the current and possible state of things, creating a world too similar to our own. Great examples like 1984 by George Orwell, The Death of Grass by John Christopher, The Day of the Triffids by John Wyndham, Soylent Green and the Mad Max Series.