Age doesn't matter in the slightest. If a case has good airflow characteristics, it doesn't matter if it's 6 weeks old or 6 years old, it'll have the same characteristics. A new case will not solve overheating issues when you have a cpu cooler that's too small for the job.
You have an old Intel lga1156 type cooler. That was back before either Intel or AMD gave a **** about your cooling solutions. As long as the cooler kept temps at under throttle while on normal loads of light Photoshop, Adobe, Office, websurfing, it was good enough. Gaming is extreme loads, not normal loads, putting a seriously higher amount of stress, voltages, heat output on the cpu. And you expect a chinsy stock cooler built for grandma's web surfer to be adequate?
- Figure out airflow. That means decent intake fan/s and decent exhaust fan/s with an appropriate fan curve setting.
- Get a cooler that's bigger and better than appropriate for the job.
You haven't said what cpu, but a minimum would be a cryorig H7, Deepcool Gammax GT, Thermalright Macho Rev.b etc.
A good cpu cooler can drop those load temps 40°C