Yeah, on the 424mkIII, and other similar Portastudios,...
you should record in HIGH speed and dbx ON.
That's the max-hifi setting: High speed/dbx ON.
The NORMAL speed and dbx OFF settings provide compatibility to normal cassettes, and backward compatibility to Tascam's "Ministudios", such as the Porta02, which run NORMAL speed, NO dbx. Also, vintage recorders like the PortaOne, ran NORMAL speed, dbx ON, so there's obviously a few different formats that Tascam wants to have covered.
The HIGH/NORMAL tape speeds and dbx ON/OFF switching gives maximum flexibility, and compatibility.
I have actually recorded normal stereo cassettes, in a normal home stereo cassette deck, and then some time later, popped them into the 424mkII, for overdubs. In this case, I ran NORMAL speed, dbx OFF.
Unless for compatibility reasons, turning dbx OFF doesn't really accomplish anything, but it lowers S/N ratio, and brings tape hiss back in, which are both undesirable.
IMO, dbx side effects are apparent only in the most rare cases, and in general, for most uses, dbx is transparent, but it definitely kills the tape hiss.
On Portastudios, unless for compatibility reasons, you should record & play back with dbx ON, at all times.