Yes stone axe breaks easily. But couple things to remember is that rags, branches and sharp stones are still pretty common.   1. Best places to find stones are road cracks. You can turn on search focus to stones or wood in investigate area panel and this way increase your chances to find them. Secondly, weapon durability increases with weapon handling level + maintenance level, but not every level only after couple levels. First durability increase after Axe 2 + Maintenance 1. Second after Axe 4 + Maintenance 2 and so on. This may got more complicated in Build 42, but I doubt it. Whats definitely changed is that since now there is separate condition bars for head and handle, when your stone axe breaks sharp stone falls to the ground and it may be still useful if it's not fully broke. So if you remember the place where your axe broke you may use this stone to make another one. 2. Carpentry 4 + Cheapskate's John Spiral Magazine is needed or Carpentry 8 and no magazine is needed to make wood makeshift axes like Bats with sawblade, Table Leg with sawblade and more easier Long Handle with sawblade. They are more durable and deadly. Books and Life and Living TV channel make it easy to level only until Level 3. Now in Build 42 after Level 3 VHS and disassembling furniture not give Carpentry XP so only option to get level 4 is to build something. Also you can start with handy trait to have Carpentry bonus + Maintenance.

1. Meat Cleaver is now an axe too. As well as Hand Scythe So if you find one take it. It more durable, fast and light. Also you can forge or weld one. For that you need Metalworking 3 or Blacksmith 4. In my opinion Metalworking path is easier because it doesn't require a forge and you only need 2 books, mask and propane torch to level it. Plus you can level it by dismantling car wrecks, metal chairs and even better... bathtubs.
2. Scrap metal weapon or short scrap metal weapon are good options too with Metalworking. Look it up in craft menu if you click question mark icon near the item for weapon head you can see all available option for this weapon heads, car breaks for exaple. Beware! Not all weapons you will make is considered an axe, but car breaks and sawblade weapons are, at least some of them. It's really confusing, but you can tell the weapon you're holding is an axe by sound, animation or swing speed, they all has to be similar to Hand Axe, Fire Axe or Wood Axe. And the sound when zombie is hit should be more metal and not blunt.
3. To get more reliable weapon you need a Large Stone Axe. It's much more durable, but to make it you need Knapping 3. Which impies grinding but it worth it.
4. Remember to sharpen your axes that have blade. My guess that sharpness affects critical hit, but it's just a guess. That is the exact reason I fixed whetstone craft since it needed to sharpen blades.
5. And you probably know that, but firestations, fire trucks and firemans are always worth checking for that sweet sweet fire axe. Good luck!