We didn't use a DFG when we had our extension built as at the time it was means tested on the parents income and we didn't qualify!!! We have however had a DFG since for the installation of a specialist bath and another ceiling track hoist. For the ceiling track hoist it was fairly straight forward. Someone from the LA came out and filled in the forms. We got 3 companies to quote - and the LA chose the cheapest - but we could have gone with the more expensive firms if we wanted and just paid the difference ourselves. It all happened quite quickly (relatively speaking in LA terms!!!).
With the one we had for the bath, we used an agency "Staying Put", who did all the planning, used a builder from their books whom they have fixed prices with, got the quotes for the bath, and arranged all the work. (They charge a fee for this, but it is included in the DFG) It took much longer to arrange than if we had organised it ourselves, but we couldn't have the bathroom out of commission for more than a couple of days, so thought it would be easier if they sorted it all, rather than us trying to co-ordinated plumbers, tilers, electricians and the specialist bath company
If you are talking major building works, then you may find that your LA will draw up plans for something you don't necessarily want, and won't be the easiest option - but may be the cheapest!!! EG. WHen we built our extension, we had an extra bedroom and bathroom put in upstairs and installed a lift, so that DD could be upstairs with us and I didn't have to go downstairs at night and through 2 rooms to get to where the garage is - which is inevitably what the LA would have converted into a bedroom if they had been doing and paying for the work.
Whatever you are planning to have done, my only advice would be, make sure you get a reputable builder - someone that has done disability adaptations before and who comes recommended.
What sort of adaptations have you got in mind OP ?