There are multiple aircraft that can operate at a higher cabin pressure. The cabin pressure is typically measured in "equivalent altitude" and ironically lower altitudes correspond to higher pressure (and vice versa).
For those who prefer "real units": 0 ft = 100 kPa, 4000 ft = 87.5 kPa, 6000 ft = 81.2 kPa, 8000 ft = 75 kPa, where kPa is a "kilo Pascal" with Pascal being the official unit of pressure (1 newton per square meter).
Legal requirement is 8000 ft and all commercial planes can do that. There are three planes that can operate at "higher pressures", i.e. 6000 ft. These are A380, A350, B787.
That means they CAN operate are higher pressures, but they don't have to and the airline is not going to guarantee it. This being said, it's a frequently advertised benefit, it's recommended by the manufacturer and it doesn't add significant operating cost, so there is little reason for the airline not to use the feature.
I was on one of the very first Lufthansa flights with the A350. Lufthansa made a big deal out of it and asked us to fill out a horribly overwrought survey to track how we were feeling during the flight.
Keep in mind that the actual pressure depends A LOT on your cruise altitude: while a clunky old Boeing 767 will can lower the pressure down to 8000 ft, it will only do so at a cruise altitude of 40,000 ft. If you are cruising at 30,000 ft the 767 operates at a pressure of about 4000 ft.
Your best bet is to go with a "modern" airplane (A380, A350, B787) and hope for the best. I also found that newer airplanes are quieter which makes IMO a big difference too. I highly recommend wearing noise cancelling headphones and ear buds. In my experience that reduces fatigue a lot more than fancy lighting or slightly low pressure and reduced fatigue makes it a lot easier to deal with ears popping and pressure problems.