Measure Coil Resistance of a Large Transformer
What am I doing wrong?

Nothing.  This is actually the expected result.  You have discovered inductance.  Congratulations*.  Had you done it about 200 years ago you'd be famous.  But you can call yourself Joe or Mike for the day with my permission.

Why can't I make a steady measurement?

You can, if you're patient.  The transformer is an inductor, which resists sudden changes in current.  Most ohmmeters are really voltmeters that put a constant current** into the device under test and then read the resulting voltage.
So when you hook up the thing, it tries to force the current through and can't -- that &quot;force&quot; is the voltage which it reads.  As you continue to force current through, the effect decays depending on the meter characteristics and the inductance of the transformer winding.
To get an accurate reading, wait a while until the inductive response fades.
As a fun experiment, do the measurement on the primary with the secondary coil open and note how long it takes to settle.  Then do it again with the secondary coil shorted.  It should take significantly less time to settle.  Repeat this while measuring the secondary, with the primary open and then shorted.
The reason this effect happens is because the short circuit is &quot;reflected&quot; back to the opposite coil, by the same mechanism that makes a transformer primary draw power when you draw power from its secondary.
* I can never tell with the written word if I'm coming off as snide or not -- I don't mean to in this case.  Inductance is pretty cool.
** As constant as they can, up to the limit of the voltage the designer is willing to put on the wires.