Shinkansen from S-Yamaguchi to S-Shimonoseki?

It's a short distance so I'm wondering if the train will make stops at both stations, and what time of day

All-stops Kodama service trains will stop at those stations. Kodama service Shinkansen run all day.

Kodama and some sakura services.

Chech hyperdia or similar timetable services for departure times.

You can search at Google MAP and HyperDia.
https://goo.gl/maps/prJLT27ZRfr
Change dep date/time and see also "SCHEDULE EXPLORER".
http://www.hyperdia.com/en/cgi/en/search.html?dep_node=SHIN-YAMAGUCHI&...
Change dep date/time at page bottom and see also "Interval timetable" of result.

In Google MAP, also may suggest "to Kokura" in Fukuoka, Kyushu region first
by "Sakura" or that will not be stopping at Shin-SHIMONOSEKI,
then back to Shin-SHIMONOSEKI by return Shinkansen "Kodama" or,
bc the operation interval of Shinkansen without transfering is one in per hour mostly.
If you have a valid Rail Pass of JR until Kokura, you can arrive earlier than using a conventional line.
(this conventional line means "JR Sanyo line", not Shinkansen tracks)

Although next month(17 MAR, Sat) JR timetables will be changing,
but in this section will not be so big, no worry.

Now, one special designed "500 TYPE EVA" shinkansen operated as "Kodama 741" in a day,
but do not know if it matches your travel date and time.
http://www.500type-eva.jp/en/
http://www.500type-eva.jp/entry/timetable.html