I think BootMagic, included as part of PartitionMagic, allows you to boot multiple DOS/Win9x OSs as does System Commander. You're going to need some kind of third party boot manager though since 9x doesn't really like to cooperate in a dual boot environment, especially with itself.
 
Although I find it overkill to dual boot 9x OSs just to get faster PS/2 port polling...
 
Have you tried a program like PS2Rate?
 
I just noticed that you have an MS mouse. MS's 2k/XP drivers allow you to change the polling rate in there, do their 9x drivers offer similar functionality?

Ever heard of Win2K? You might install it and when you find you love it you can simply single boot a real OS . If you're worried about some of your applications not working you check for compatibility records right here at NT Compatible. WinME sucks and Microsoft knows it's a discontinued platform so don't expect a lot of product support on that one. By the way is this a secondary computer or something? Otherwise what are you doing at NT Compatible if you're using 9X only and don't have plans to move to a stable platform any time soon?
-Christian

If you have 2 hard disks or use PartitionMagic to create 2 active boot partitions you can install 98 on one and Me on the other. However, when booting into the other OS you must mark the other OS's partition as hidden. Only 1 can be active at a time.
 
Simply put just pick either ME or 98, 98 is nearly dead, ME while it isn't the best of the 9x OSes, it'll be supported (games and stuff) for at least as long as Windows 2000 is around.

I agree with Christianb 9.X (95-ME) is dead, like DOS. It should only be run on sub 300MHz computers. Dualbooting ME and 98 is like dualbooting DOS 6.2 and DOS 6.22. It's not a stupid question. Go with NT 5.x you'll wish you had a long time ago.