M.P.
I strongly suggest that you spend as much time being with her and holding her as you possibly can. She is feeling very insecure. You've made a major move to a different country. Everything around her his different. She's very anxious and perhaps even frightened, especially if you without intending to but because of your feelings of being overwhelmed are pushing her away.
This move is a major change for you too. I suggest that you spend the next few weeks focusing on supporting her and each other. Pick her up and hold her even when she's not asking for it. Give her lots of attention.
And most of all, find a way to manage your feelings so that you aren't feeling irritated and drained. Perhaps you and your husband can spell each other so that one of you can get out of the house and do things for your self. You need time without her so that you can be all there for her when you're with her.
Because she was a cheerful, well adjust toddler before she'll be back to that soon. In the meantime she's only asking from you what she needs. Think of her as being anxious and needing reassurance. Give her all the attention that you can. When you're feeling anxious and overwhelmed pass her along to your husband and take a break for yourself. Work out a routine with him so that you can deal with your toddler. She'll probably let him hold her if you're not there unless he's more stoic and less able to soothe her. If that's the case, leave her with him anyway. You need the break.
She will be less whiney and demanding if you're able to cheerfully give her the attention she needs. Hold her when she's not whining. Work on being proactive rather than reactive to her needs.
I agree, this is not the time for a playgroup unless you're able to stay with her while she plays.
This too shall pass.