OK, so in DA, if I pull the trigger to rear to about 90% of a full trigger pull, and let of the trigger, the sear will trip. It will be as if I pulled the DA trigger 100%. If there was a round in the chamber, it would fire. It does not do this with the slide off.
In SA, if I can push the trigger forward of its normal SA position. Enough so that there is a tactile and audible click.
I can try and post a video, but it will be a day or two.
90% of the normal DA trigger travel?!? Why are you pulling the trigger in DA that far and trying to "let off"? That's a very good way to experience a ND with a loaded gun.
If you pull the trigger in DA mode far enough (which is before the trigger reaches and bottoms out against the frame at the end of its possible travel arc), the drawbar's throw notches are going to cam off the hammer's corresponding DA throw notches and the hammer is going to be released to fall forward. (Sooner, if the throw notches are worn, damaged or there's some other problem.)
The check for "rocking the trigger in DA", which is done with the slide off the frame, doesn't go back nearly that far. Maybe halfway, at the most. Easier to see it demonstrated, and practiced, in a class than to discuss it in text. Armorer students learn right away that if they pull the trigger too far in DA, the hammer is going to fall. (Save that for the other test, confirming the hammer falls in DA as it's supposed to fall when the trigger is fully pulled to rear.

)
In SA mode ...
If the
trigger can be moved forward a little bit when the hammer/trigger are cocked in SA, a small amount of movement is considered normal. That's "trigger play" which may occur before the trigger hooks engage the drawbar and push it forward to release the hammer. The trigger play spring usually helps to keep that sort of "slop" from occurring, or at least being noticeable.
When the hammer is cocked into SA - with the hammer's SA notches resting on the sear's nose - the trigger's forward "hooks" are positioned forward within the generous area of the drawbar head's V-notch. The points of the "hooks" (or ears or prongs, if you'd rather) may move a bit within the tall and wide drawbar head V-notch.
The movement of the hook's points can manifest at the longer "other end" of the trigger assembly as the potential for a minor back/forth movement. Hence, the role of the trigger play spring in tensioning the trigger in SA. The gun will function normally without the trigger play spring (and some LE guns have been ordered without them installed), but the trigger can have a little bit of loose "wiggle" (slop) in the SA mode.
When the trigger is intentionally pulled in SA, though, the points of the trigger's hooks drop forward into the center of the drawbar head's V-notch and push the drawbar forward. This causes the leading inside edge of the drawbar's tail to contact the rear of the sear and push the top of the sear to pivot forward, pushing the sear nose out from under the hammer's SA notches. SA hammer fall.
Sometimes the tips of the trigger hooks (especially in the .45/10 frames) might end up rising high enough so the they push up just over the top edge of the drawbar head. (Outside the top of the V-notch). Pulling the trigger can result in a "clicking" sound as the points of the hooks drop down over the top edge and go back into the V-notch. Annoying to some owners, but not considered an actual functional "problem" according to the folks at S&W.
Here's a pic I took of the optimal relationship of a trigger play spring holding the trigger hooks forward in the center of the V-notch. This was just done to show the angle of the spring under tension against the rear of the trigger hooks, with the assemblies outside the frame of a gun.
With and without a trigger play spring on the drawbars, to "simulate" the function of the spring to help reduce trigger movement in SA mode. Since we're talking about a pivoting trigger ... if the
top of the trigger, the hooks, drops back a bit rearward, that's going to translate into forward movement of the bottom of the trigger.
Trigger hooks may vary a bit, within normal tolerance spec (length, sharpness, roughness of edges, etc), among both machined and MIM triggers. (The flash-chromed trigger marked DA is for an older DAO 3rd gen.)
Here's 3 different drawbars (a couple different calibers/widths), showing the V-notch from another side angle.
Wish I could be there to handle your 4506, and perhaps answer your questions. I'm just a former S&W armorer, though, not a gunsmith. Perhaps you should call around your area and find a gunsmith who is familiar with S&W 3rd gen TDA (DA/SA) guns, to inspect yours to see if everything is in normal, good working condition.