How Should I Handle this 'Cousin' of His?

Good evening sister Amara. I thank God for what he has been doing through you. I have been following you for more than three years now, please  l need godly counsel.

I am married for close to seven years now, though it has been a distant marriage but I finally joined my husband few months ago. My husband has a female cousin who is very close to him, he even told me how close they were when we were dating. But on joining my husband, I realised they relate almost like real lovers.
They call themselves all sorts of pet names and romantic texts.

I confronted him but he told me that there's nothing between them, that they are blood related. My major concern and fear is that he calls her secretly, and sends her money secretly.

After a lot of quarrel about her, I told him to abate my fears that he should be calling her in my presence but he bluntly refused, that it's because am jealous.

One day, unknown to him that I was coming back, I was at the door overhearing his conversation on phone with her. He was asking her if her husband doesn't see their text, also that he's more stronger in keeping their vow. My heart sank, I couldn't hold myself, I knocked n he cut d line immediately. I asked him who he was talking with but he denied even talking on phone.

I have nagged, begged, quarrelled, done everything  I could but he still refused to control thier relationship. Please I need advice on what to do. Thank you.

First thing, investigate on the identity of this 'cousin' of his and find out how related they are to each other. When you have ascertained that and you are pretty much convinced that they are genuinely related to each other, then you will need to transform your nagging, begging and quarreling to productive decision. 

Marriage is built on salient terms and conditions, and there are things you shouldn't manage in your marriage; cheating is one of them.

Cousins don't keep vows, and cousins don't have an intimate relationship that is strong enough to be kept in the dark from the significant other.  This cousin is a strange one, and you need to address it with every sense of respect, maturity, and wisdom.

Discuss with him on clear terms how hurting this is to you, then engage someone who he genuinely respects and share your concerns with him/her. 

This person must be an elder/family relatives or a cleric or a counselor, and he/she must be an unbiased individual who can tell the truth without compromise. 

What you know is what you heard and read, but what you don't know is what they've shared with each other. Praying alone won't solve this, it's either he's choosing to stick with you by making amends or you decide whether the marriage is safe enough to you or not. 
Pretending to be married when you are dying slowly is the worst deception you can do to yourself. If he refuses to make amends and give priority to his marriage, then you may need to consider separation to regain your sanity and peace of mind.