There's a few minor things wrong with this installation, but I'm not convinced any of them being corrected will make much difference, maybe all together may make a difference?
It is always a good idea to avoid unnecessary bends in any drainage system, so the disposer outlet should go out then straight down through a 90 ell, preferably a long turn ell, into the P-trap. No other inlets or outlets here, just one pipe closed pipe into the P-trap.
There is no need for a vent on the inlet side of the trap, there should be one in the wall from the original P-trap connection. I'm assuming the new P-trap is connected to the vertical pipe used by the old P-trap, yes? If not and not vented, this is a problem that needs to be addressed. Every P-trap must have a vent on the outlet side.
The dishwasher should drain into the disposer through the small connector above the outlet. Some dishwashers do not adequately grind food waste, draining through the disposer ensures it gets done one way or another. If there is not a physical air gap fitting for the dishwasher, the flex pipe should at a minimum be attached to the underside of the counter to prevent drain back into the dishwasher from the disposer. If the disposer's dishwasher connection has never been used, there is a knock out plug inside the fitting that needs to be removed. Be sure to retrieve the plug from inside the disposer before running it.
This will give you a completely proper installation and hopefully eliminate your clogging issue.
Update
This sketch should better illustrate what I'm suggesting.
Once again, this is assuming the old trap was properly vented in the wall and the new connection ties into the vertical waste pipe of the old trap. This will mean the new trap should also be properly vented. If not properly vented, the trap will not function properly.
Update 2
Answering additional questions. Yes, venting is the big unknown. If the old trap worked well and you never noticed wastewater odors coming out of the sink drain, the venting is probably correct, especially if the new trap is tied into the same vertical waste pipe. If the new line went off somewhere else I'd be more concerned. It looks to me like you tied into a former cleanout formed by a vertical tee, in which case I am extremely confident all is well.
The air gap is required by some codes to make it impossible for sink waste to get into the dishwasher. This is important because of the potable water connection in the dishwasher. There is a slight chance of cross contamination without an air gap. If the waste pipe is merely fastened close to the underside of the countertop, this achieves a very similar effect with a high point, though cross contamination possibility goes from impossible to highly improbable. If the waste pipe were to go straight into the disposer without a air gap or high point, the cross contamination possibility becomes real. Even though the chance is still slight, the fact it is possible is something that for me is disgusting to even think about. Old food bits in drinking water? Ewww!
Pipe F. No, do not remove this. Hard to tell from the photo, but it appears you can just rotate the U trap portion anti-clockwise so that it's inlet is aligned directly below the ell outlet. By rotating the U portion and/or the disposer itself, you can achieve any sort of alignment required. Only the length of the outlet pipe from the disposer would need actual cutting (and the vertical pipe into the trap inlet of course). The horizontal pipe I drew going into the wall is not required, I didn't have your photo up for reference when I did the sketch, which I see now was a mistake. Sorry. Don't do that, leave that portion as it is. As long as the fittings connect together as shown, the actual lengths of pipe in between, or if there is no pipe at all, none of that matters.
Best Answer
Yes use one for both the waste and vent. If the existing sink doesn't have issues draining, your vent will be fine. For the waste, just "T" the two sinks together and join them to the existing drain/waste pipe. Do the same for the water supply for the sink. Remove the existing cut-off valves and put a tee on the stubbed out pipe with the opposite end of the tee pointing at you and the leg of the tee pointed to the new sink. Run a new piece of pipe from the tee over to your new sink and put a 90 on the end. Connect your shut-offs for the old sink at the tee and for the new sink at the 90. Of course you'll have to run a hot and cold line. You don't have to, but I would replace the existing cut-offs with new ones while they are off. Good luck!
Normally you would stub out a drain for each sink and then have a p-trap directly under each. This is what is shown in the photo , before the p-traps are installed. It is permissible to have one trap, but if you do, it has to go below the tee. I'm old school and don't use Shark fittings. I'm not sure if they are acceptable in a sealed wall or not. Hopefully someone else can help you there.