There is a trick to the scroll saw.
You need to have the right combo of:
1- blade
2- wood thickness
3- speed.
If the current speed sort of scares you and you are worried about getting your fingers near the blade, then slow it down to where it is comfortable. It will lessen the vibrations and give you better control.
If you have a giant thick blade on thin wood then that combo is off. Imagine you are cutting veneer with a big old saw from the garage, it will splinter and break and make a mess.
Now if you had a fine toothed saw and the right amount of pressure and speed then it will look great.
There is a chart around here somewhere that explains the right sized blades for the wood you are working with. if you cant find it here just goolge it.