How do you handle being clingy in a relationship?

Hey gals! Alright, so as the title says, How do you handle being clingy in a relationship?
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I'm sure there are girls out there who have went through this problem and overcame it and remained a healthy relationship. Please feel free to even give advice if you're single, from previous relationships and etc.!
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Him and I have been together for about a month now and so far no problems, however I feel like I'm going through that falling in love phase. I've never been this clingy before and it's getting to my emotions. He does work a lot and he's really busy but I just want to stay with him. Please give advice and help me out, thanks girl! x