Hi Am
There are so many errors, I am limited by the Forum Rules on the level of help I can give for a Credits project assignment.
Think of a way to use blocking/steering diodes on the two LM324 Outputs so that they are not linked together.
Also, the Trig input of the 555 requires a capacitor coupling.
I skipped from page 1 to page 3, so this might have been covered already --
You cannot tie the outputs of two LM324 sections together directly. These are totem-pole outputs, and will attempt to create a dead short between V+ and V-. I have no idea how various simulation programs would handle this, but it probably is incorrect.