How to troubleshoot engine vibration?

I have a 1998 Ford Mustang, 3.8l. I swapped the motor for the same 3.8l. Everything is all buttoned up. The motor starts and runs. Its idles normal in park and in neutral. Once it is put into any gear, the motor shakes a bit, has a low idle, and a severe loss of power. Taking the car on the street is not an option because of the lack of power. Almost like a major vacuum leak. I checked and replaced all vacuum lines, new plugs and wires, new mass airflow sensor, all fluids are full, fan turns on and thermostat is working great, no leaks. OBD2 with no check engine light. Can't seem to figure out what is causing this issue. I was thinking the harmonic balancer, but can't seem to find anything wrong with it. Any suggestions as to what I should check next?

Added (1). I used the o2 sensors from the old motor, the harness from the new motor. I used the old throttle body, iac, altenator, battery, and pulleys. The car is running rich, the exhaust fumes are too much to bare while the car is running in the garage with the door opened fully. I had one check engine light, mass air flow sensor, replaced it, and the light was gone. I hooked the scanner up again and there were no codes stored. I ran the diagnostics and found my o2 sensors running at 0.00

Added (2). Also, I used the old trans, torque converter and flexplate from the old motor

You stated that you replaced the engine but didn't say whether you used all of your sensors, wiring harnesses, etc from your existing engine or did the new engine come with its own sensors, etc? Even if the check engine light isn't on, that doesn't necessarily mean there isn't a code stored. I would do a compression test first so you can make sure the pistons, valves, etc are ok.A burned valve, sticking valve, or worn cam lobe could cause the symptoms you listed. If the egr valve came with the engine, I would make sure its functioning properly and if its the type of egr with passages on the backside, remove it and use carb cleaner to clean all excess carbon out of those passages. The fact that the symptoms only appear when you place it in gear seems to point more towards a sensor or component problem vs mechanical. I assume the correct balance flexplate is bein used too. The balancer thought you brought up could be looked into also.

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