Technically, one can accept Shabbos early, but must still wait to say sefiras haomer until it has become full night. The reason is that we start Shabbos earlier than the time that the new day has started as far as the halachos of sefirah is concerned. Thus, during the sefirah period, we actually do not say the sefiras haomer in shul. It is similar to repeating the krias shmah after the appropriate zman when one has gone home.
In fact, one can still count for Thursday after davening Friday night and accepting Shabbos, while saying the count for Friday with a bracha after the correct time has passed.
The reason that this is done, is that Sefira is based on the 24 hour period, starting after nightfall.
Your mention of waiting to start Ma'ariv for Shavuos, is because once Shavuos has started, the Sefirah period has been ended. Thus, one must wait until one would have been able to count day 50 having completed all 49 days in order for Shavuos to start.
Catching Up On Sefirah On Shabbat Eve
Question: What if someone forgot to count sefirah Thursday evening but only realized after he finished davening Friday evening? The catch
is that he accepted Shabbos early so that it is still light outside.
Can he still count for Thursday evening and then count for Friday
night with a berachah once it gets dark?
The short answer to your question is that a person may still count the
Omer for Thursday evening after Friday night davening if it is still
light outside and continue counting that night and subsequent nights
with a berachah (HaGaon Rabbi Moshe Feinstein, Responsa Iggrot Moshe,
Vol. 6; Orach Chayyim, Part 4, 99:3).
The halacha is similar to a niddah counting seven clean days.
The Taz writes that Sefirat HaOmer serves as a proof for the Agur’s
view since we know that even though tosefet Shabbat – adding time to
Shabbat – is a biblical concept, we can’t count Friday evening’s
sefirah until it is actually dark.
We see that the counting of the Omer is comparable to the counting of
the niddah, and what applies to one therefore applies to the other (in
regards to the counting itself, not the blessing). The actual time for
counting in both situations begins at nightfall.