I think my husband is lying about what he does after work, but I don’t know how to feel about it

My husband is a manager at a restaurant and he always works closing shifts. His GM said that he doesn’t want him at work past 1 o’clock in the morning if he can help it and to get his stuff done before then. My husband comes home around 230 to 3 usually every night though and he doesn’t work that far away. He lied to me or as he says “omitted information” about going to hang out with his friend a couple weeks ago. He had this big elaborate story that he tried to call me, tried to text me, his car wouldn’t start, etc. So the next day I go look on my phone and I have no make up missed calls no missed texts he has no calls to me or text me. So we’re driving in my van to his car and there’s a text from his friend saying that he moved his car, meaning my husbands car, to a different part of the parking lot. So I’m wondering hear how his friend moved his car when he said he couldn’t get it to start And that’s why his friend had to drive him home. Well, turns out that he and his friend went out for drinks after work while I was sleeping and I’m assuming he met his friend at his work and they both took his friends car to go out. My has been was probably too drunk to be able to drive so that’s why his friend brought him home. So I talk to about all of this And he gets all defensive and start arguing with me about how omitting information Is not lying. But eventually, he does confess to what really happened. The other night, kind of the same scenario only he didn’t come home until after I called him at 5:30 in the morning. He says that his car died yet again, he couldn’t find his cord for his phone so it died, he tried to call his boss and then he Ended up falling asleep in his car. I called him for probably 20 minutes over and over, text him, video called him. Finally he answers the phone, and tells me all the stuff that happened and that 40 minutes before I called he had to walk in go get a cord from the store to be able to charge his phone. I’m sitting here thinking huh... How did you charge your phone from your dead car that is miraculously working now and you’re on your way home? There is no transaction record from the card that he said he used to buy the cord with on our bank app. Before all of this stuff, things just keep happening at work that make him stay at work longer. I don’t know what to do with it. I can’t just keep turning a blind eye and let him get away with whatever it is he’s doing but he denies any wrongdoing on his part and just expects me to believe that he’s telling me the truth. He says that he wants a relationship to be good in everything but yet he doesn’t seem to care enough to prove to me that he is actually doing what he says he’s been doing. I wouldn’t have such a problem with this except that we are going through marriage counseling trying to get our relationship better and it’s really hard for me to deal with this when we have both committed ourselves to make things better between us. Our kids deserve the best from us. What do you all think I should do? I love him and care for him deeply but I can’t just let him run all over me and expect me to always be there for him when he’s not being honest with me. We haven’t been to the counselor since these things have happened in the last couple weeks. I intend to talk to him about these things with my husband present and see what he thinks. Thanks for listening, I don’t have very many friends to be able to talk about this stuff with. I also apologize for any grammar errors. I am voice texting on my phone.