Your problem certainly is transmission related. I would narrow your choices down to these:
- Torque converter
- Clutches
- Bands
- Gearing
If it looks like this is a broad range of choices, you are right. Physically, at least, you are looking at four different, but related, areas of your transmission. The thing of it is this, they are all in the same acceleration chain, if you will, that enables your Toyota to accelerate from a start to 50 kph and back down again to 0 kph. The torque converter takes the power from the input shaft and transfers it through the valve body to the clutches and bands that are activated by the solenoids. The clutches and bands then, working with the gearing, provide your Toyota with each gear in cycle.
Now, it is possible that the input shaft or the shaft seals may be out of round and vibrating which will set the entire acceleration chain off kilter. It’s possible that this is what is causing your vibration, but I think not. I think that when you removed the old transmission fluid that was black and burnt various deposits were freed up by the new transmission fluid — the stuff that became blackened and burnt quickly — and they floated through the transmission case, only to become lodged somewhere else. Most likely, the new deposits filled in various valve spaces in the valve body so that, at key moments, when the transmission fluid was supposed to be opening a check valve to flow through to the new gear settings needed by the clutches, bands and gears to operate correctly, there was no fluid.
In turn, the lack of flowing transmission fluid or smoothly flowing transmission fluid damaged the clutch/band combination and/or the clutch/band/gear combination, causing the problems you are having.
If you want to hold onto your Toyota, which I suspect you do, the only thing for it is a rebuilt transmission to replace the one that is now in the case. Though it depends on your physical location, you are looking at an investment of up to $3,800 (U.S.). I know that is probably not the answer you want to hear, but it is, nevertheless, the only answer that is possible.