Advice needed: Overbearingly clingy guy won't let me not talk to him

So I'm really into this guy. He's sweet and caring, understands mental illness, and really romantic. I love being with him, but the problem is when I'm NOT with him. Yesterday while I was at work, he accused me of lying to him when I wasn't replying. I work in an assisted living facility and when I wasn't replying, I happened to be serving dinner and the cook doesn't allow phones in the kitchen. He claimed I told him I was out at 3:30, not 7. When I replied and said "I never said I was out at 3:30, I said I had a BREAK then" he still said I was lying, so I asked if it would make him feel better if I sent copies of my schedule to him, and he replied saying "wow thanks for making me feel like an ass".

This morning I woke up feeling particularly introverted and not wanting to talk, so I sent him a message saying so. Here's the following convo

I really like him and want to be with him, but I have no idea how to get beyond this. Sometimes I just don't wanna talk, and sending emojis back and forth constantly for hours is annoying af. I don't know what to do, so any advice you could give is greatly appreciated.