I'd use a piece of chain and a hook, so when you stick it into gear, it will hold the handle in place. Maybe several hooks in seperate places on the chain for other gears. OR---You should check the handle detent for wear on the "teeth"or slots that hold the shifter handle in the speed you want to use. Also, check for something under the machine that may have come loose.
There are other web-sites that you can use for info on yer problem, if ya want to look, or Google them.

Better advice is to hook the chain to the shifting rod, to hold it up. Due to age, it's probably made of softer steel. Then insert the rod into the deck hole.

Have you considered the possibility that a mount or torque strap could be loose on the transaxle?

Is this a manual trans or hydro?

314H = Hydro
314-8 = manual

There is a method to adjust the friction of the disk that holds the joystick steady. On my 312, it's under the seat. It's pretty self-explanatory. You tighten a nut and it makes it increasingly hard to move the joystick.