Hello; Once you get the issue of adding some additional support to take the load off of the screws settled, a way to address the wobble is to add a sheet of plywood to the back of the frame. You could also add sheets to the ends sort of like making a partial box. This should prevent the racking (wobble). The plywood sheets should not need to be to be thick, 3/8's should be enough to stiffen the stand.
A quick way to add some strength might be to drill some holes thru the legs and add some carrage type bolts. I would also put large flat washers on the bolts.
Find out what sort of screws were used. If they turn out to be drywall screws, that can be an issue. Such screws are reported to be very brittle and will snap (Shear) in place of bending. If such screws were used I would get some stronger rated screws and replace them one at a time and perhaps add a few more.
More pictures at various angles might give us better information. It may be that you can salvage the current stand.
Good luck