Nikki, you can make that bench a lot less shaky very easily.
If you can get two pieces of even thin metal strapping, or wood as small as 1" x 2", and make an 'X' shape that covers the back, even just the lower section, and with 5 small bolts attach it at the back, one bolt for each leg of the 'X', one bolt for the crossover of the 'X', it will make it much less flexible.
Same for each side if it wiggles front to back.
An easy fix, world of difference.