Part difference - regulator 50kHz flutter
The datasheet for the LF33CDT shows that the part is from before 2004.  Revision 14 of the datasheet is in 2004, so it was around for a long time before that.
The age is relevant.  The datasheet consistently shows electrolytic capacitors for the output capacitor of every example.
At that time, those would have been tantalum electrolytic capacitors.
The datasheet for the LDF33DT is from 2013.  It consistently shows nonpolarized capacitors as output capacitors.
Those will be ceramic capacitors.
There lies the difference.
Ceramic capacitors have much lower equivalent series resistance (ESR) than tantalum capacitors.
The LF33 needs that higher ESR.  Without it, it will oscillate.
Your circuit was designed for the LDF33 and uses ceramic capacitors.  The LF33 is oscillating because the ESR is too low.
You can either replace the ceramic output capacitor with a tantalum part with proper ratings, or you can add a few ohms of resistance in series with the ceramic capacitor.
Adding a resistor means either changing the layout or bodging the resistor in to the existing space.
Switching to tantalum capacitors means having to be prepared for the tantalum &quot;vent with flame&quot; failure mode.

A third option would of course be to use a regulator with the same pinout that would &quot;be happy&quot; with a ceramic output capacitor.
The NCP5501DT33RKG suggested in the comments has charts showing stable operation areas in relation to output current and ESR.
The ESR values shown are probably only achievable with ceramic capacitors.  In addition, the example circuits all show nonpolarized output capacitors.  That makes it very likely that ceramic capacitors are expected.
From that point of view, the NCP5501DT33RKG should work in your circuit.
Aside from that, it looks like at least some of the housing styles are compatible with the LDF33DT.
It ought to work.  No guarantees, though.  That's your job.  :)