GO! Yes, adjusting to a radically different climate and country will be challenging. But your children (and you) will learn SO much! As for managing logistics with 3 kids and a hubby who is gone long hours - I've only visited there, but have had several friends who have lived in Singapore, and here's what I know:
1. Housing is expensive. Depending on how much housing $$ comes with the job, you might be in an apartment. But apartment living is the norm in Singapore, and can be very comfortable. And since nearly everyone does it, apartment buildings tend to be much more family-friendly than they are in the U.S.
2. Domestic help is (or at least, was 10 years ago) dirt cheap. My friends who have lived there on middle class $$ have managed to hire both housecleaning and childcare help.
3. Private vehicles are expensive. Public transportation is plentiful, efficient, and reasonable.
4. There is an extensive international community there. There are lots of Western expats. A key to not feeling isolated is to tap into that community as quickly as possible. Join a community rec center or church, since you are homeschooling and won't have school connections. Perhaps there is an expat homeschooling group? The permanent population is an interesting mix of ethnic Chinese, Malay, Indian, and some Europeans. English and Chinese are the dominant languages.
5. As a homeschooling mom, what a treasure trove you will have at your fingertips! History, natural sciences, geography, language, culture, art... you will be living in one of the richest cultural crossroads on the planet. Also, if you leave the city part of Singapore, there are tropical forest nature preserves and beaches to explore. If you drive across to the Asian mainland, there are gorgeous beaches and world-class scuba sites. Don't do that alone with your kids, though - Singapore is quite safe. Malaysia, not quite so much for an obviously foreign woman.
I've lived abroad (Japan, Korea) and loved it. My first child was born in Korea, so I've been abroad with a newborn, in a much less Westerner-friendly culture than Singapore. One thing I've noticed about Asian cultures in general is that children are cherished and generally looked out for in that part of the world. I've had people who wouldn't have given me the time of day hold doors for me and give up seats for the baby. Kids in Korea cross streets without fear - cars would stop for them. They wouldn't stop for me.
In fact, the friendliness toward kids may take a little getting used to - adults will talk to your kids all the time, and might offer them candy or small toys. You'll need to think ahead about how you'll deal with that. I elected to be friendly back, and if my kids were the same ages as yours, I think I'd coach them ahead on how to be polite and that accepting gifts from strangers is only okay if you're with a parent, and that once the giver has moved on the gift goes into mama's hands. (Koreans liked to give my baby hard candy - well-meant but misguided.) Older ladies may ask to hold your baby. I allowed it (I figured if I needed to, I could take out a rogue grandma).
It is worth a long conversation with your husband about salary, living allowances, housing allowances, and possibly school $$$ - now that I think about it, I do not know what the homeschooling rules are in Singapore, or how much leeway expats are allowed. I do know that 10 years ago the international school there had an excellent reputation. (I hope that you are able to homeschool, though - who would want to be stuck in a classroom with all that adventure waiting outside the door?) It is also worth seeing if you can talk to someone from the company who is living over there right now or has returned recently, preferably someone with kids. That way you can get a better feel for what the reality on the ground is.
Would you be giving birth in the U.S. or in Singapore? Don't let the idea of giving birth abroad be a deal-breaker. Yes, it is different, but that doesn't necessarily make it worse. Natural childbirth is the norm in much of Asia, but then again, so is having midwives on staff at hospitals. Research birthing practices in Singapore if you're worried. Personally, I had a better birthing experience in Korea than I did here, and the entire birth - prenatal care, hospital, doctor, midwife - cost me $2000 before insurance, in a hospital that has received UN recognition for its work with women and children. The facilities were basic and institutional, and I did pay extra for a private room. (Most Koreans give birth in a communal birthing room. I wasn't ready to go THAT local.) The people and their skills were top-notch. If necessary, you could always remain behind with your parents and give birth in the U.S. before heading over. Your husband could handle finding a place to live and moving in on his own.