You should NOT use 1 5/8" screws for standard 1/2" drywall work. There are 2 reasons.
First is that building codes require that things like supply pipes and electrical wires are installed in holes in the center of a stud. A stud is 3 1/2" wide. Let's say there is a 3/4" pipe running through the center of that stud. Allowing for 1/8" room to fit on all sides, there would be a 1" hole drilled in the middle of the stud for it. That leaves 1 1/4" of wood on either side of that hole. A 1 1/4" screw would penetrate 3/4 of an inch, leaving a half inch margin for error. A 1 5/8" screw would penetrate 1 1/8". Now you have 1/8" margin for error. And if you've ever seen pipes and electrical wires installed in your studs, you KNOW they were not that precise. Of course there should be nailing plates installed, but there are often not these either. You significantly increase your risk of puncturing something with 1 5/8" screws.
The second reason is that not only are longer screws not needed to hold up drywall, they can actually do more harm than good. To take it to the extreme, imagine a 4" long screw, which would penetrate the full 3 1/2" into the stud. Now, any warping or twisting of that stud over time is sure to create a "nail pop" so to speak (I think you know what I'm talking about). This actually weakens the drywall installation. You want just enough screw to hold firmly, but no more so there is less risk of nail pops when the studs commonly move slightly over time.
USG recommends 1 1/4" drywall screws ONLY for drywall up to 5/8".
"On 3/8", 1/2" and 5/8" thick panels, use 1-1/4" Type W Bugle Head Screws for superior holding power and high resistance to popping due to wood shrinkage."
(Hopefully it would go without saying, but also never use fine threaded (i.e. metal stud) screws for wood studs. I see this all the time - it's too weak. The "type W" referred to above refers to Wood type or coarse threaded screws.)