Hello,
Since the adapter worked fine at the shop, the hardware is probably good to go and the issue is just between the laptop and your specific home router.
Does that white X show up before you even try to type in the Wi-Fi password, or does it happen right after it tries and fails to connect? Also, are your phone or other devices connecting to your home Wi-Fi normally right now? Sometimes routers are set to block new connections automatically until you allow them.
Get back to me with those details and we will figure out our next move.