on the premise that at this point a few pictures might be in order, I dug my just pulled 350-E out of its current corner and took some close up of the various identifying aspects.
Here it can be clearly seen that the ATF pan has its passenger corner clipped to create an angle to the oil pan. As the thread above commented this is what he found when he managed to slither his way under the car to look. Also visible in the picture is the vacuum module c/w its vacuum line that would be attached at the other end to a fitting on the manifold if this unit was in a vehicle.
These show what typically would be seen if looking at driver's side of the E or lock up version of the 350. The first pic is nothing unusual, just the gear selector lever.
PIC Two shows the external clue that this transmission is not a normal or standard version. This is the plugin into the wiring harness shown in the next picture would be attached. This is power gets to the transmission to initiate the lockup of the converter in third gear. Visible are the four sockets in the female body but if you look closely you will only see one pin in the male fitting. Not sure why but suspect that what was actually happening was that the lockup occurred due to input from other sources and this is the circuit that is attached to the brake pedal to break or shut down the lockup during braking or declerating.
To answer your question about the need for the lockup. Yes, you can live without it; the e-version of the 350 will happily move from gear to gear without issue and exactly like its non lockup cousin would do. The concern is internal heat that the transmission can generate while going about its business. If you surf sites that deal with transmissions you will discover that transmissions possessing the lockup capability but having it disabled will run hotter and require more frequent ATF changes.
Okay, this being the passenger side of the 350-E what you see is nothing unusual or different between it and its non-zapped cousin. The open port showing the red ATF is where the dipstick tube plugs in. Both pics also show the demand downshift cable, aka the passing gear cable location with the cable still installed.
Finally a detail shot of the carburetor end of the downshift cable. This one has a custom created extension/adapter attached to it to allow it to be fitted to a Holley 4 Barrel that I used to run. I went EFI a few years ago but needed to keep the adapter at the time. My new 700R4 doesn't need it so it will come off this cable and get parked (and probably eventually lost.)
So about the only visual cue that would tip you to having the E variant is the plug in on the transmission case. As I noted, apparently as far as the 350 goes you can live without the lockup. I did with this unit for over thirty years but most of that was stop and go commuter traffic. Had I been running the highways or interstates it likely would have failed years ago. By the way, this unit was not original to the vehicle it came from, Its predecessor blew its guts out and all over the concrete pad at my service station. Good thing they were an understanding and sympathetic bunch. The box was done; a Total DOA. Lost the entire contents of the pan. T-mission shop owner took one look at the corpse and turned and walked over to his shelves to see if he had a rebuilt replacement. Lucky for me he did.
One thing to note here. I cannot honestly state that the first transmission blew up because i had been running it on the interstate with the lockup disabled. The vehicle itself was a used unit and I got a "good" deal on it because they didn't want to do the regular maintenance.
I can say that the replacement unit never saw any interstate time.
Most of the time what the lockup will do is add a few miles to your miles per gallon and take some of the stress off the drivetrain in the process.
Good luck with your project and welcome to the forums.
Nick