Hi Lulu. Yours is such a difficult situation isn't it? I speak from my own experience because I can understand as me and my hubby are from different countries and we live in England, but at the beginning when we got married, 16 years ago, we used to live on his home-country and I felt really odd, despite people and his family being nice to me. Over here, I feel odd sometimes but I do feel like kind of my second home.
In my opinion sex is very important in a relationship, we can't denial it is not only for the physical aspect but mainly for the intimacy and feeling of really being together between a couple. Without it things turn weird. But reading your post it feels like ok you guys haven't had sex for a while, but it seems there are more issues that are getting in the way: you're always busy and with loads to do, your OH always grumpy and tired and don't making an effort to appreciate you, kids to care for, the stress of daily life basically.
For my own experience again, I haven't had sex with my hubby for ages (longer than you) and we feel that's not right. Slowly, we started to communicate and he's aware that I've sex toys and indulge in solo play and he's ok with that. The main fact for us to stop having sex is that I felt he didn't appreciate me as I was the one always dressing up, putting lights, candles, lovely music on and he enjoyed, but kind of went through the motions and foreplay for me is essential, even more so than sex. So I thought: "ok no foreplay, no giving me something back and time, sex is out the cards". But because now we're communicating, we feel like slowly a door is opening for us, we are getting closer and it's nice being together again, he's showing an interest in my sex toys and what I do with that, we are nearly there, but it's taking time, years to be honest. Without talking, patience and work this wouldn't have happened.
All families have problems and that saying "the grass is always greener on the other side of the fence" is true, you don't really know if those families are that happy, but because you feel so upset right now it looks like they are. When we are out as family, I'm sure everyone thinks that we are very happy because we laugh a lot, my hubby for all his wrongs, got a wicked sense of humour and I love that. But hey-ho our life is a nightmare as bringing up teens is a challenge, it drive us insane I tell you.
Like RoseCheek suggested, counselling might be very good for you and your OH as you need feel like he listens to you and care enough to at least try do something. I know you're busy Lulu but perhaps you should find something that gives you as a person (not a mum, a wife or a professional) some kind of pleasure, like yoga, massage, gym, medidation, masturbation etc. It's essential you look after yourself as well because, again my own experience, no OH, no kids, no-one can do that, just you. Find something you like and stick to it because that's part of looking after your own health and well-being and that would give you energy and inner strenght to deal with the challenges of your relationship and life right now. However busy your life is, you need to find time for yourself.
Children pick on our feelings all the time, specially on the "bad" vibes and you're right it's not good for them, but it's not good for you pretend everything is ok for their benefit. Tell your OH how you feel and the reasons you don't feel like having sex right now. Perhaps he too has reasons to feel fed-up, grumpy and tired all the time. If you guys don't try to open the lines of communication, you'll end up resenting each other and things getting even worse. And only you can honestly say whether it's worth or not. You know sometimes despite everything, it's not mean to be.
I hope you can work thing outs Lulu and I honestly wish you good luck. Keep us posted.