Amy Duggar King & Jill Duggar Dillard Lean on Each Other After Joseph's Arrest

Amy Duggar King has gotten pretty close with her cousin Jill Duggar Dillard ... especially in the wake of Joseph Duggar's arrest.

In a new YouTube chat with interviewer Dory Jackson ... Amy says she and Jill -- who's one of Joseph and Josh Duggar's sisters -- have been texting a ton since Joseph was arrested in March for allegedly sexually abusing a minor in Florida.

Amy said ... "We have definitely sent messages back and forth, all week long, you know, for weeks now of just like, 'Are you doing OK? I love you.'"

She notes their bond used to be "surface-level" ... but it's much deeper now. Amy explained ... "We always knew each other, fun and happy and cousins and all that, but we never really talked about heart-deep issues and the things that we might have trouble with or the things that we might struggle with or ... trust."

She says they're practicing developing that trust, adding ... "You know, we’re not going to blab what we're told and what we talk about. And, you know, that's what makes a friendship so close [or] a cousinship so close is that family really relies on each other."

As you know ... Joseph was arrested and charged with lewd and lascivious conduct with a minor. His arrest comes almost five years after his brother Josh was found guilty on child pornography charges.