This is probably only a partial answer, but comments don't allow many words.
The Nikon term TTL has three meanings, interpreted from context.
One is just the generic meaning automatic flash mode. The menus generally mean that in most all cases. The menu saying TTL means whatever TTL Mode the camera is going to do.
Another use is that film TTL mode (automatic) was called TTL.
Todays digital version is called iTTL. But in some cases, the term TTL means explicitly "film automatic flash mode", but not likely in your modern case.
Another use is specific to Nikon automatic flash modes. The Nikon camera flash metering has always had two modes, called TTL and TTL BL.
TTL BL is and was the default, and is Balanced Flash. In bright light, the TTL BL mode is compensated to be a weaker fill level, to better match the strong ambient.
TTL mode is the old standard automatic mode, the flash is metered independently and regardless of ambient. The TTL flash comes ahead on strong regardless of how bright the ambient might be (and overexposure is the likely result in bright ambient, unless the user compensates it weaker.) Regular old TTL, but Nikon default has been TTL BL for quite many years.
In dim ambient, there really should not be any difference in TTL and TTL BL.
But bright ambient will reduce the flash level to be fill level, called Balanced.
The flash menu says TTL on or off camera.
If on camera, the camera reports its default mode is TTL BL, but is the report, it still says TTL on the menu.
The SB-700 has no selection menu for TTL or TTL BL, but the default camera metering mode is TTL BL. The only way to switch to TTL instead is to select camera Spot Metering. The flash system NEVER does Spot metering (camera ambient does), but Spot Metering switches the flash metering from default TTL BL to be TTL.
All of this was all exactly the same on the D3200 and SB-700, and much older too, it has not changed in ten years.
Why your flash appears weak may be due to Auto ISO. Don't use Auto ISO with flash, set your ISO 800 directly, NOT Auto.
Nikon has changed the way Auto ISO works with flash a few times.
Older early iTTL models would not increase Auto ISO if a flash was detected present. That was great (because, we are using flash instead).
Then about D300S era, this changed, and Auto ISO depended ONLY on the ambient metering, and the flash had to work into the high ISO it found in effect.
That was poor, but we could turn Auto ISO off.
Then around D800 era, they stopped that, and if flash was present, Auto ISO would ONLY increase at most two stops in Auto ISO mode (two stops above whatever Minimum ISO said). Bounce often needs the two stops to 400.
I believe your D3200 worked this current way, and surely the D5600 does too.
So pay attention to your ISO, and your case may go better with flash if you simply turn Auto ISO off, and set ISO 800 directly. Or maybe set 800 as Minimum ISO if you actually want Auto ISO, but Auto ISO is largely not effective with flash.
If you are using camera mode Auto (as opposed to A,S,P, or M), then Auto ISO is on (Auto Everything is On), you cannot turn it off in Auto mode.
What are your actual camera mode and flash modes?
So why weak flash?
I assume you are NOT using Commander mode in any way, not on flash and not on camera?
Make sure your ISO is what you think it is. ISO will show on the rear LCD after every picture. The up/down buttons select that screen that shows settings for the previous picture.
Make sure you don't have Flash Compensation or Exposure Compensation at some unknown value, affecting you. The top LCD (by shutter button) has a black +/- icon symbol that will be on if any compensation is set anywhere. So if symbol shows, find out what it is, and fix it.