A plot of the Moon-rise/sets on the Horizon Location We have been trying to see where the moon rises/sets on the horizon.   Unfortunately, the moon is not cooperating.   Even when we know the moon rise/set times (timeanddate.com), we often cannot find the moon on the horizon.
We are wondering if the moon wobbles back and forth along the horizon the same way the sun wobbles.  We cannot find any website that talks about where the moon is on the horizon.
We are wondering if the moon wobbles back and forth along the horizon the same way the sun wobbles.

Yes, but over a period of 27 days compared to a full year for the Sun.
If you visualize a compass, the angle that a star rises and sets is the same angle measured from due south (for those in the northern hemisphere). This figure shows a star rising 60 degrees from due south (at an azimuth of 120 degrees). The same star sets 60 degrees from due south (at an azimuth of 240 degrees).

The rise angle and set angle can be calculated from
$$cos(angle) = -\frac{sin(dec)}{cos(lat)}$$
where $dec$ is the declination of the star and $lat$ is the observer's latitude.
The same calculation applies reasonably well to the Sun and Moon. (The apparent diameters makes a small difference, as does atmospheric refraction. Let's ignore those effects for now.) Due to the motion of the Moon during the day (and to a smaller extent the Sun), the change in declination changes the rise and set angles by a small amount - a few degrees at most. For the following graphs, I have ignored this motion during the day.
For the year 2021, you can see how the rise/set angle and azimuth of the Moon changes from &quot;north of east/west&quot; to &quot;south of east/west&quot; during one month (approximately). Of course, the Sun requires an entire year to go through the same cycle.

The azimuth changes because the declination changes, and the declination changes because the Earth's axis is tilted 23.5 degrees from perpendicular to the orbital plane. The Moon's orbit is inclined another 5 to 6 degrees from that. As Uhoh has mentioned, the inclination of the Moon's orbit sometimes adds to the declination and sometimes subtracts from it. This longer cycle takes 18.6 years to complete. The rise/set angle for a full cycle looks as follows:

Here is another factor that may be affecting your observations. Between New Moon and Full, the Moon rises during the day. It may be difficult to see the Moonrise until it gets close to full. During this time span, the Moonset occurs at night. (So &quot;measure&quot; the Moonset and calculate the set angle from due south.) The opposite occurs from Full Moon to New: it rises during the night and sets during the day.