Thanks for "micbooth" explanation.
2134 is nice and stable OP yet not too expensive, something I can recommend for almost every situation. Go for it.
2227 is nice, prolly better than 2134 at some aspects, but its too laid back and fatty to my taste. I haven't got a chance to use 2107, but some people swear on it. Definitely something worth checking for. Some Ops from AD are very nice too. They usually are little hard to use though. As you already know, each OP produces sort of unique sound with different sound signature. So... you know... you keep that in mind, and choose OP that you like the most within your budget.
So... go back to cheap 4556....
Yup, its jellybean kind of cheap OP chip.
However there're several jellybean OPs that is not too bad at all. 4556 is a one of them.
I had used a handful of them at bunch of Cmoys, A47s and several other OP based amps. Used them at both gain and buffer stages, and at power rail position too. Never had any problem with them. That makes me consider this cheap OP as a descent alternative of more expensive OPs, especially when I think little thin and hollow, yet acceptable sound of 4556 is suitable for what I am making. If it gets hot in your amp, other amp may get hot too at the same circuit. That means something is not right. Could be the gain or grouding issue. I think you have to take a good look at your circuit and power.
BTW, what headphones you're connectiong to A47?