You should never have to feel that your current relationship is being dictated by your X. However, if your X is likely to have objections, you would be foolish to develop a new relationship unless you have well-thought-through plans in place as to how you are going to handle those objections - both in terms of what you'll verbally to your X, as well as in practical terms about maintaining those boundaries, i.e. do not plan dates or weekends for times when your X has contact unless you have a babysitter on standby.
Failure to eliminate problems caused by your X will lead to a lot of fallout. Not only will your resentment towards your X colour your ability to effectively co-parent with him, but the fallout on your new relationship could easily lead to your new P feeling you don't stand up to your X and you feeling you're caught between a rock and a hard place, the kids not knowing what to feel or who to feel loyalty towards... you get the picture.
I'm slightly worried by the fact that your new relationship was 'slightly rocky' to begin with. In its initial flush should be the time when problems are minimal and certainly seem surmountable. If neither of you were really quite ready for a new relationship when you first started dating, I'd make sure you take things very, very slowly indeed, even though those problems now seem to be resolved.
In terms of introducing the DC. I see no reason not to. If your new P and your DC can't get on, the relationship has no future. Better to find out now how that's going to work. There is nothing wrong with early introductions, contrary to popular belief, it's all about how it's done. DC aren't stupid either. They know the difference between a friend and a 'boyfriend' so you may as well tell them straight up.
I'd get into the habit of allowing your P in for a quick coffee before you go out on dates. And just leave it like that for a good long while. Don't have him stay overnight in your bed (that's very significant for DC), don't have him do anything that's traditionally parent territory (e.g. bathing, putting to bed). Avoid day-to-day close family/parent type actions such as school reading book tasks until your DC have well-and-truly accepted his presence. In essence, just take it slowly and introduce a little bit more at a time. Problems start when it's rushed (usually for adult convenience), the irony being that it is exactly this that causes the most problems for the adults further down the line.
I took well over a year introducing DP to my DC, from the first time he had coffee to the first time he read them a bed-time story. He didn't stay overnight until 6 months after he'd met them,and only then occasionally, not regularly. It was nearly 2 years before he was another member of the household and nearly 3 before he moved in completely. That's probably longer than most people would consider and I'm not saying my way is the only way that works. However, I think it worked because my DC never felt they were losing me to him, rather they felt he was an addition to the adults who cared about them. And we've never had any of the problems that can sometimes rear their heads in step-families.
If he's worth it, he'll go along with whatever you want.
Good luck.