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.