3V Current source with low voltage drop

The LM2596 is probably a bit better than what you think. If you look at figure 8 in the data sheet: -

It tells you that for a 1A load (at 25ºC) the drop-out is about 0.9V. Also note that the output is now just slightly out of regulation (Vout = Vreg - 50mV).

If you are running less than 1A this figure will improve.

However, take note of figure 11: -

It is telling you that at 25ºC the minimum operating supply voltage is about 3.6V (and this id for producing a miserly 1.23V on the output).

There are better devices than this I suspect and you need to consider what your maximum load current is when selecting one.

The LP2950 is a linear regulator and if you look at figure 10 it tells you that for a 50 ohm load and 4V inputted you will get about 3.6V coming out. Also look at figure 12 because this gives a different viewpoint in terms of output current.

Yes it can be used as a constant current generator. With reference to the circuit: -

Take all the ground connections and group them together - this is now going to be the output voltage point for your load and the voltage across R2 (ADJ pin) will cause current regulation when this voltage is 1.24 volts. In other words if R2 is 10 ohms then 0.24 amps flowing into your load will be the current limit. R1 may be chosen to be quite low but the data sheet doesn't appear to say if this can be zero ohms. Maybe an experiment or more digging around will find an answer.

This method isn't limited to this regulator. Here's a 7805 regulator turned current source: -