First, I'd say...take it slow. You've only been in the picture fulltime for a few months, yet you are talking about them calling you mom, adoption, etc. Blended families take a LOT of time to...well...blend!
Background: I'm a custodial step mom, and have been for 5 years. My kids see their mom a little less than yours do - each kid has gone from 9 months - 1 year without seeing mom at all. We bonded VERY quickly, but even then, there's been ups and down.
I don't remember if you mentioned how old the kids are. But, most step parents will tell you that it's a long process, and is often the "one step forward, two steps back" process.
The first thing that jumped out is the 'mommy' thing. Don't expect, and don't push them, to call you mom or mommy. They have a mom, and she is in the picture, although not much. It's very important for them to maintain that relationship, no matter how screwed up she might be (i'm guessing about that - of course, she could be lovely). So, if you don't like them calling you by your first name, pick something else that's neutral - even "Mommy K." can be loaded. In my case, I've done all the mom things for these kids for 5 years, and they now even routinely put me down in the "mother's information" section of paperwork. But I still don't expect them to call me mom. That's reserved for their mom. If mom totally disappears, then all bets are off, but if she maintains any contact, decide together something to call you that makes you both comfortable.
Expect that they will talk about their mom, especially after seeing her. Also, don't be surprised if they idealize her - 'mom would see us more if she could'. They are trying to make things work in their head, and face it, few kids want to admit that their mom doesn't want to see them! Never ever say anything bad about their mom to them, even if she says something horrible about you -kids WILL remember who said the bad things and who doesn't (LOTS of experience on this one!)
I'm not sure about the adoption statement - is BM (birth mom) willing to sign over her parental rights? If not, I think adoption is out of the question, isn't it?
Yes, step kids will push your buttons - so do your birth kids! Keep in mind that you will have all the challenges of an intact birth family (kids pushing buttons, testing limits/authority) AND all the challenges of a blended family.
Sounds like you are pushing hard to make everything a nice neat package, but the reality is, it rarely works that way. Let things find their own level. There will be some days when you love each other like nothing else,and some days when the kids (or you!) will want and need some space. That's perfectly normal, and things will eventually find their own level. Good luck!