Thanks for providing more details about your preferences and style of play. I see you have a good understanding of the game.
The fact that you play more backhand and more straight drives than crosscourts is great, because it shows that you don't struggle too much with things like racquet preparation, swing, and follow-through, and your footwork must be pretty good too.
Let's see if we can figure out why you keep over hitting the ball. The technique and the targets on the front wall obviously play a big part in it.
The easiest solution could be to aim your shots a bit lower on the front wall, but it's not always possible or appropriate.
Also, maybe you're bringing the racquet too high during your swing preparation. A shorter swing could help improve accuracy.
Another tip: just be steady when you hit. Stop, then hit. It can be done almost simultaneously. I know It's not so easy as we don't have the luxury of time to prepare for a shot like in golf :)
It's also possible that your racquet face angle is too open. Tweak that to hit the ball flatter, which won't be easy to master but it's a great skill to have in your arsenal. You can pound the ball pretty hard when hitting flat and the ball will have good length most of the time.
You can think of a flat shot as a topspin shot, except you shouldn't add any spin, but the racquet face angle is rather similar, so at least that should give you some idea. Start your swing like you're going to do a top spin, but turn the racquet face flatter. If you accidentally add some topspin, don’t worry. It might actually work in your favour and what was supposed to be good length may turn into a winner. That said, topspin is one of the most advanced shots in squash and the outcome may be quite volatile without a proper skill set. And lastly, make sure to hit the ball at the top of the bounce or when it slightly drops, but you can't hold your shot for too long or you'd lose the right angle.
Also, cutting the ball can reduce over-hitting but it's tough to hit good length consistently.
By the way, you said that you prefer thicker grips. I'd recommend trying thinner grips which might help you develop a better touch and control.
Now, let's talk a bit about the racquet. You mentioned that you prefer a head heavy balance. Similar to tennis, squash racquets with a head-heavy balance tend to have more power and less control.
As you master your technique, you usually don't want your racquet to be too head heavy. It’s just hard to get a good feel, not to mention the lack of manoeuvrability.
The downside of the head light racquet though is that it becomes quite weak when you hit it at the top, especially between 11 and 1 o'clock.
This is to say that you should master your good touch before switching to a more head light racquet.
Based on your preferences, I think that our new PXT Incognito might be a good fit.
With the full strung weight of about 142-145 g, it's almost as light as it gets and the balance point of 360 mm is by 5 mm-10 mm more head light than many other high-end racquets on the market.
It's definitely an advanced racquet, but I think it can help you learn how to hit shots with good length more consistently.