OK - I have experience here, and i am happy to pass it on.
First of all - remember this - it's the most important fact all parents need to know -
kids will always play one parent off against the other given the chance. It's not nasty or malicious or scheming, it's the way children are as they explore relationships and how they fit into the world. All kids do it with all parents, the problems get bigger when one is a step-parent for all the reasons you have stated.
Once kids get wind of the fact that you are tiptoing because of anyone's opinion of what you do - you are doomed to be a doormat for ever.
So - you need to have a chat with your OH and get some ground rules betwen you.
You must agree to back each other up at ALL times, no matter if one of you thinks the other is wrong, infront of the children, it's a united front - discuss it in private if either of you has issues.
Your partner tells the children that at your home, you are 'mum' in all but name, with the same rights and discipline sanctions - and he supports you in that.
This is to get things ready for the 'bad times'.
You do your best to show the kids that they are loved and respected in return - goes a long way to heading off the bad behaviour born out of insecurity and fear of loss.
When they play up, your approach is "You are making me unhappy with your behaviour, and i don;t want to be unhappy because i love you ..." which will work until escalation time - which will come, and then you need punishment snactions which you carry through as warned EVERY TIME, and appealiong to Dad has NO EFFECT whatever, so they soon give up trying.
Lastly - he will tell their mum, as firmly as necessary, that this is your relationship, and the kids will behave in your home, and neither of you will be listening to any tale-telling from the children about anything at all. Start from a zero