Your local Social Services should have the forms, and, hopefully a competent social worker to help you fill them in.
You do not need a Solicitor, but if you do, ask what their experience is in this field !
The Court will send an officer called a Guardian ad Litem to asses your application and the reasons for it. (I was one).
One reason people consider adoption in this situation is to change the name of the children,
This does not need adoption, just a simple notice to all who have contact with them, school doctor, Benefit providers regarding allowances etc. Deed Pollis not needed at this time.
Another reason is "to prevent any further acces from the natural father". Courts are not happy with this, and feel a child needs to know, and have, if possible, contact with their father. ( who will have to consent remember, and will be interviewed).
In any case the Birth Certificate WILL say "By Adoption" and you will need to prepare them for this, as they will certainly see this when older.
I Too, am sorry to be a wet blanket, I hope all goes well, but better to think deeply now than have tears later.