When I'm in command, should I also do an oath of office when promoting someone from WO1 to CW2?

The technically correct answer is that no new oath is required (10 U.S. Code § 578(f)*) unless the individual had a break in service as a warrant officer. The Oath of Office given to all officers, warrant and commissioned, is the same one regardless if they are being appointed as a WO1, CW2, 2LT or COL (5 U.S. Code § 3331*).

However, there is absolutely nothing wrong with administering an Oath of Office as part of the ceremony when someone is promoted and it is very common to do so (e.g., I was given a new oath of office on three of my five appointments to a higher grade) - it just isn't required.
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* 10 U.S. Code § 578 - (Warrant Officers) Promotions: how made; effective date - https://www.law.cornell.edu/uscode/text/10/578
* 5 U.S. Code § 3331 - Oath of office - https://www.law.cornell.edu/uscode/text/5/3331

When I was commissioned, I took a new oath of office. I don't know what the rules are now, but I would guess that at some point, somebody will ask him where that is in his file.

Yes, he should have done an Oath of Office and you'll want to too.

I would. Better Safe than sorry! COL Randall Cudworth LTC Trent Klug Lt Col Charlie Brown COL Mikel J. Burroughs

There is no requirement to do so. It is not uncommon. Just as it isn't uncommon for Marine SNCO's to recite the Oath of Enlistment when promoted to the next rank as a sign of commitment to the Marine Corps and their Marines. But there is no requirement forcing them to do so or forcing you to advise your Commander otherwise. Be careful not to overstep. But ultimately for this setting it falls into preference.

2LT(P) (Join to see)
OH Ya Mean: "The Commander-In-Chief of All American Military Services"?
- While Stationed At Osan AFB, South Korea, I Happen To Know The Kim Family And "Fa"
Was The Head Of the Household, And What Fa Said, Goes. - So I'd Take His Advice When
Referring To Our: "The Commander-In-Chief of All American Military Services" So Rest Assure
"Fa-Kim" Says it Best.

I Can See No Reason It Shouldn't Be.
Is That Not A Part Of His Responsibilities?

title 10 part 2 chapter 36 sec 626b. is it a requirement, no as there was no break in service.
is it traditionally done as part of the ceremony yes.