No don't push harder, that's how you break ****!!!
The problem is caused by the clutch dragging and nothing else. You'll probably all notice that as it gets hotter, it gets worse. I worked like hell to fix it on mine and got close but it's impossible to get rid of the dragging completely, nature of the beast I guess. The things i've found that help:
When using reverse, throw that lever back into forward drive while you're still rolling. You can't damage anything by doing it, and it gears will mesh together much smoother when they're actually turning. When you are stopping and want to hit neutral, try hitting it while you're coasting to a stop and the gears are meshing. Just keeping those techniques in mind should be able to make the problem survivable for most of you.
Avoid idling for long periods of time or riding slowly when possible, these are air cooled engines so they're going to get hotter if you're going slow or stopped, and heat = thermal expansion = clutch & transmission dragging.
Good oil. If you use crap oil, the transmission will work like crap, the clutch will drag, etc. Synthetic seems to be best but at least a decent brand dino oil will help (maxima, amsoil, silkolene, belray etc.) Valvoline and Yamalube were the worst for me. Not only does the oil lubricate the clutch and components, but it also is responsible for keeping it cool, and as parts get hot they expand and everything gets tight and drags.
Make sure the cable is in good condition and adjusted properly, so that it is fully disengaging the clutch. Try tightening the cable and see if it alleviates the problem at all.
As already stated, a good direct shifter can sometimes help finding neutral, but if the clutch is dragging badly, it's not going to do much.
Try replacing the clutch if the problem is really bothersome. Go with stock Yamaha parts, and pay attention to the parts that don't usually come with clutch kits. Those wavy seperator rings are there to help seperate the clutch pads when it's disengaged, and over time they will flatten out and become useless. Check the basket fingers for wear and file/sand them down smoothly or replace if necessary. Those grooves will make the clutch hang = dragging. Check all the bushings, throwout bearing, etc. and replace if doubtful about them being in good condition.
The oil cooler mods also seem to help as long as the problem isn't being caused by warped clutch plates or other damaged/worn components. It will still have limited effectiveness at low speeds because it still needs air moving to work well. I actually installed 2 120mm fans on my oil cooler and then later added a cooling fan from a 660 blowing directly at the head of the engine. I put the fans on a manual switch so that I could turn them on only when I needed them, running constantly that 660 fan would probably start drawing down the battery.