Was Umbridge still technically the DADA teacher after becoming High Inquisitor?

Question

I was looking up the Jinx on the DADA position and I thought, "Why did Umbridge fall victim to its effects if she was High Inquisitor, rather than DADA teacher, by the end of the school year?" Then I realised she could very well have been both.

However, whilst looking into it, I noticed that summaries of the story of Order of the Phoenix rather gloss over whether or not this is the case. After becoming High Inquisitor, she begins to throw out decrees left and right, and instead of DADA classes, we get to see / read about Dumbledore's Army meetups instead, which of course are far more entertaining. But perhaps I missed something.

Is there anything to confirm or deny whether Dolores Umbridge continued teaching Defence Against the Dark Arts after becoming High Inquisitor at Hogwarts, such as if she discontinued the class entirely, or had some lackey teaching the classes for her, but keeping to the same useless curriculum?

Answer 1

Yes.

She becomes High Inquisitor in Chapter 15:

MINISTRY SEEKS EDUCATIONAL REFORM: DOLORES UMBRIDGE APPOINTED FIRST EVER HIGH INQUISITOR
_{Order of the Phoenix: Chapter 15 - The Hogwarts High Inquisitor}

And later in the same chapter we see mention of her still teaching DADA:

But Harry did not have to wait for Defence Against the Dark Arts to meet Professor Umbridge.
_{Order of the Phoenix: Chapter 15 - The Hogwarts High Inquisitor}

And again:

All the while, Professor Umbridge stood a few feet away, making notes on that clipboard, and when the bell rang she descended the silver ladder first and was waiting for them all when they reached their Defence Against the Dark Arts lesson ten minutes later.
_{Order of the Phoenix: Chapter 15 - The Hogwarts High Inquisitor}

And she says she is still the DADA teacher

‘I was Senior Undersecretary to the Minister, yes,’ said Umbridge, now pacing around the cabin, taking in every tiny detail within, from the haversack against the wall to the abandoned travelling cloak. ‘I am now the Defence Against the Dark Arts teacher’
_{Order of the Phoenix: Chapter 20 - Hagrid’s Tale}

She becomes Headmistress in Chapter 28:

BY ORDER OF THE MINISTRY OF MAGIC
Dolores Jane Umbridge (High Inquisitor) has replaced Albus Dumbledore as Head of Hogwarts School of Witchcraft and Wizardry.
The above is in accordance with Educational Decree Number Twenty-eight.
Signed: Cornelius Oswald Fudge, Minister for Magic.
_{Order of the Phoenix: Chapter 28 - Snape’s Worst Memory}

Then remains in charge of DADA:

‘Oh, no need, thank you, Minerva,’ simpered Professor Umbridge, who had just coughed her loudest yet. ‘I was just concerned that you might not have Harry’s most recent Defence Against the Dark Arts marks in front of you. I’m quite sure I slipped in a note.’
_{Order of the Phoenix: Chapter 29 - Careers Advice}

And actually teaches DADA:

Professor Umbridge was still breathing as though she had just run a race when she strode into their Defence Against the Dark Arts lesson that afternoon.
_{Order of the Phoenix: Chapter 29 - Careers Advice}

So yes. Umbridge remains the DADA teacher after becoming the High inquisitor and even after becoming Head Mistress.

Answer 2

There is even more explicit evidence than that mentioned in the above answers. The Daily Prophet article that mentions the appointment of the High Inquisitor clearly states (my emphasis):

Professor Umbridge has been offered this position in addition to her own teaching post, and we are delighted to say that she has accepted.’

Answer 3

Yes, she was

As there was no official announcement of another DADA teacher, and the appointment was to last one school year, we can deduce that Dolores Umbridge has not resigned this post. Therefore she was continuing to be a DADA teacher, combining this post with that of a Headmaster and High Inquisitor.