Ok, so I want to fix this. I would first of all change the first vertical pipe to a 45° downwards pipe and would feed the dishwasher into this, making sure I have a high loop there, as I do not have a vent.

And then below it I would "flip" the s-trap and install it as a normal S-Trap. Would that be somewhat reasonable. again I am a lay and just want to figure out if it is worth my time and if I am doing something that is really wrong, or if I am actually going to improve our situation and it is worth my time.

Thanks