Welcome to SO.
Your problem might be somewhere else. ST_Distance with two geometries using the SRS 32613 returns the distance in metres:
SELECT ST_Distance('SRID=32613;POINT(508654.55672303465 4390740.143711988)',
'SRID=32613;POINT(508654.55672303480 4390740.143711988)');
st_distance
----------------------
1.74622982740402e-10
(1 row)
It also works using ST_Transform
SELECT ST_Distance(
ST_Transform('SRID=4326;POINT(-104.89910 39.66643)',32613),
ST_Transform('SRID=4326;POINT(-104.89907 39.66643)',32613));
st_distance
------------------
2.57321026907276
(1 row)
Demo: db<>fiddle
Are you perhaps mixing the order of the coordinate pairs? Remember, it is longitude, latitude, not the other way around. If the geometries are correct, please post a WKT literal from both geometries, so that we can reproduce your environment. Another option would be to use geography instead of geometry, which would automatically return the result in metres, but you would need to transform the geometries encoded in 32613 in a lon/lat coordinate system to make the cast work, such as 4326.
EDIT: Read carefully the answer of @JGH - he might have found the real issue. You're probably using the coordinates with a wrong SRS!