Does the Holy Spirit always indwell the believers?

Yes

“And I will pray the Father, and He will give you another Helper, that He may abide with you forever-–”— John 14:16 (NKJV)

The New Covenant or Testament is unlike the Old Testament when the Holy Spirit left the believers.

Samson

“And she said, "The Philistines are upon you, Samson!" So he awoke from his sleep, and said, "I will go out as before, at other times, and shake myself free!" But he did not know that the Lord had departed from him.” Judg 16:20 (NKJV)

Saul

“But the Spirit of the Lord departed from Saul, and a distressing spirit from the Lord troubled him.” 1Samaul 16:14 (NKJV)

David feared the Departure of the Spirit from him

“Do not cast me away from Your presence, And do not take Your Holy Spirit from me.” Ps 51:11 (NKJV)

There are times it might feel like he has departed but be sure he is still there , if you grieve him , or queched him that could be the reason but he hasn't left.

“And do not grieve the Holy Spirit of God, by whom you were sealed for the day of redemption.— Eph 4:30 (NKJV)

“Do not quench the Spirit.”1Ths 5:19 (NKJV)

You still remain his temple .

“Do you not know that you are the temple of God and that the Spirit of God dwells in you?” 1 Corinthians 3:16 (NKJV)

But that doesn't mean he won't chantes you , for his discipline is strong and sometimes it could lead to physical death.

“If anyone defiles the temple of God, God will destroy him. For the temple of God is holy, which temple you are.” 1 Corinthians 3:17 (NKJV)

1 Corinthians 11 (NKJV)²⁹ For he who eats and drinks in an unworthy manner eats and drinks judgment to himself, not discerning the Lord's body.³⁰ For this reason many are weak and sick among you, and many sleep.

“Then Ananias, hearing these words, fell down and breathed his last. So great fear came upon all those who heard these things.— Acts 5:5 (NKJV)

Acts 5 (NKJV)⁹ Then Peter said to her, "How is it that you have agreed together to test the Spirit of the Lord? Look, the feet of those who have buried your husband are at the door, and they will carry you out."¹⁰ Then immediately she fell down at his feet and breathed her last. And the young men came in and found her dead, and carrying her out, buried her by her husband.

Holy Spirit will never leave us and is our down payment for something more grand the indwelling of the Father and Son in the Believers.

2 Corinthians 5:5 LEB Now the one who has prepared us for this very thing is God, who has given us the down payment, the Spirit.

Ephesians 1 (NKJV)¹³ In Him you also trusted, after you heard the word of truth, the gospel of your salvation; in whom also, having believed, you were sealed with the Holy Spirit of promise,¹⁴ who is the guarantee of our inheritance until the redemption of the purchased possession, to the praise of His glory.

“Jesus answered and said to him, "If anyone loves Me, he will keep My word; and My Father will love him, and We will come to him and make Our home with him.”John 14:23 (NKJV)

“To whom God would make known what is the riches of the glory of this mystery among the Gentiles; which is Christ in you, the hope of glory:”— Colossians 1:27 (KJV)

For those who state that but Christ already dwells in us , he does throu the Holy Spirit but his physical presence does not , hope in "Hope of glory" implies it hasn't happened yet and we are waiting for it and so does the scripture teach us the meaning of hope.

Romans 8 (NKJV)²⁴ For we were saved in this hope, but hope that is seen is not hope; for why does one still hope for what he sees?²⁵ But if we hope for what we do not see, we eagerly wait for it with perseverance.

The Christain is to have the Three in one God dwelling in them Father, Son and Holy Spirit.