What mirrajay said!
To even think about starting this process you will need your husband to consent to terminate his parental rights(TPR)-
even if you currently have sole custody and he doesn't see/support your children. My stepfather could not adopt me because my bio father would not consent to TPR though he was incapable of taking care of me. Because we resided in different states the courts would not decree TPR. I would imagine this to be impossible from abroad.
Are your kids currently in the UK? (I assume yes from your question). If they are still in the US- your case will be considered an Inter-country adoption- which are damn near impossible and take twice (or more) as long.
Step family adoption within the UK is just as lengthy and expensive as non family adoption, it can take around a year for the most straight forward cases to go through. There is no guarantee that your DH would be your children's legal guardian by the time you need to apply for ILR.
The first step in the process (after securing TPR) would be to contact your local council's social services. They will set up a date for a social worker to contact you about putting forward an intention to adopt.
After this is completed it will be handled like a normal adoption, with background checks, interviews, home studies. This can take 6 months to a year, if started on time, some councils have back-logged cases so it can take longer. When these checks are over you will be given consent to put forth your petition for legal adoption and the case will be brought forward in front of a judge.
I have no direct experience with stepfamily adoption in the UK, but I have heard that it doesn't always get granted because here the courts are of the notion- unless the child is in direct danger they believe parental rights should always belong to the biological parent. I can say in the US this is exactly what happened to me, and my bio-father was considered unfit to care for a child.
IMHO, it will most likely be cheaper and simpler to get your children ILR.
Good luck!
Also, now that I typed this and just found that direct gov has some really good basic info covering what I just said:
http://www.direct.gov.uk/en/Parents/Adoptionfosteringandchildrenincare/AdoptionAndFostering/DG_10021340