^ Things can be rough in an environment like yours but if any tuner can do it I'd say it is the Sony. Part of what makes the Sony so good is that although it is very sensitive and can pick up a weak signal, it curiously is also very selective
and can ignore nearby strong signals on the dial or even the same frequency but coming from a rival station in a different direction in another city/state:
"Capture ratio is how far below an unmodulated 65-dBf monophonic signal a 100%-modulated monophonic signal must be to obtain the specified quieting. With both signals in stereo, the XDR-F1HD can suppress a co-channel signal 19 dB stronger than one the HDT-1X can suppress. At my location this profoundly improves reception quality for many signals, setting the Sony apart from any other tuner.The astronomical selectivity figure is real.
The XDR-F1HD is noticeably more selective than the HDT-1X, sometimes retrieving listenable signals that are buried beneath adjacent-channel splatter or completely inaudible in the Sangean. The Sony's selectivity is more than 30 dB better than that of the best conventional tuner I've ever tested, a Kenwood L-07TII modified to cascade one 150-kHz and two 110-kHz Murata ceramic filters in narrow-IF mode." [emphasis mine]
- from one of the reviews I've lost the link for but I'm sure anyone could search for the original source if need be.
Of course there's only so far this can take you and if your local transmitters are right on top of you the tuner just can't hear anything else because it is being deafened by their loud shouting, as you already know. I'd recommend the unit to you but if it still is having issues I'd recommend three changes:
A) Chuck the supplied dipole antenna which is nearly omni-directional and get a directional antenna like a "Yagi" you can aim away from the overly strong stations as much as possible and towards the hard to get ones. This must be used in conjunction with
B) an in line variable RF attenuator, suitable for FM not just TV (no built-in FM "trap"), mounted right at the antenna input to reduce the strength of the antenna's signal. Remember the Sony has good sensitivity so an intentionally weaken signal can still be great but the attenuation kills the overwhelming, problem signal.
C) Buy or build a Faraday cage to put the tuner in. (Sine the Sony comes with an IR remote you should be able to operate most features even with the thing trapped inside a metal mesh or wire box. This will ensure that the circuits are only getting an RF input from the antenna terminals and the various solder points inside the tuner aren't accidentally acting as their own omnidirectional antennas which would be swamped by those too strong stations you have nearby.
edit to add: I have not worked with them personally, but some say one can use their unplugged microwave oven as an impromptu Faraday cage box for test purposes. Of course you won't be able to shut the door fully with the wires to the tuner hanging out the front, but it may be a good way to see if the shielding it provides will stop the part of the tuner that is being overwhelmed from being overwhelmed, and if the results are good then building or buying a real
Faraday cage should be more likely to solve your problem.
Sorry if what I'm suggesting might seem a little nutty.