My question involves adoption law for the State of: Alabama

My husband and I were awarded custody of his 2 kids in October 2010. They are now 5 & 3. We filed for custody after hearing from his ex that she was in an abusive relationship that resulted in several 911 calls, hospital visits, etc. She was asked by the judge if she wanted her kids or her boyfriend. She stood up & said "I can't leave him because I might be expecting" so the judge gave the kids to us, sole physical & legal. We had the kids 5.5 out of 6 months before that anyway. We've been back & forth to court every 3 months since then with the judge giving her chances to straighten up. She was told to move out of the hotel she was living in, get a full time job and stay away from the boyfriend. So far she has moved at least 11 times but has done none of the other things. The boyfriend is in jail & she visits him regularly. She is also on her 3rd "fiance" since October 2010. She started taking college classes but quit 3 months ago. She supposedly works 5-8hrs/week but we've seen no proof of that.
We did not ask for child support because frankly we knew it would never get paid. She relies on her aunt, whatever man she is with at the time & various churches for money.
Since we were awarded custody, she rarely calls. She went 4 months without laying eyes on them last year. We recently moved closer to her but she still has only seen them twice in 4 months. She never asks to see them. It was always me calling her but I've stopped doing that. I'm not forcing her to spend time with them. The kids don't even ask to anymore. We saw her a few days ago in public, the kids wouldn't even speak to her. The 3yr old just clung to my leg like he was terrified.
In February 2012 she lost her visitation rights due to an incident with her oldest son, that is not my husband's child. That child was given to her aunt. Both cases against her (abuse/neglect)are due back in court on the 17th of this month.
The 5 yr old has sever ADHD among other things. I've been responsible for them since April 2010. I do everything...feed, clothe, bathe, dr visits, therapy appts. I take the 5yr old to all his dr/therapy appts for ADHD. I take the 3yr old to his asthma specialist. The bio-mom can't even name their doctors or the meds they take. I potty trained both of them. They are with me 24/7. I quit my job to be with these children as they really needed a full time mother.
The ex has stated to me more than once that she only wants them back for food stamps & TANF benefits. I have an email where she offered us $500/month in food stamps if we would just give her physical custody so she can file them on her case. She wanted the kids to stay with us but wanted court documents to say they lived with her.
I would really like to adopt these boys. I feel like she would gladly sign for me to adopt if we still let her see them periodically but it's just not good for them. The way things are looking, she will most likely not be gaining back her visitation rights on the 17th.
Can I just file for adoption without letting her know? I mean, I realize she will be served papers to sign & that she will have to agree to it but can I just file without telling her...letting her find out when she receives the paperwork? This going back & forth to court is ridiculous and it really is not benefiting the kids. I just want whats best for them. We've been the only stability in their life for 2yrs now.
Sorry this got so long. I'm actually leaving out a lot of details on bio-mom because her list of bad parenting skills & life choices is a mile long. Anyway, any/all advice will be greatly appreciated! Thanks!