1. I'm not sure why you are thinking of stopping the ESA claim. If you are in the "work related activity group" it does not necessarily mean you are expected to look for work straight away. Rather, you can be expected to enter into actions which can lead to you being ready to find work at a later date. Meanwhile, you get about £93 per week ESA.
2. I suggest you appeal to try to get into the Support Group, but do not end your claim. Co-operate as much as you can. If you can't get into the Support Group you can decide then whether you can cope with continuing in the Work Related Group.
3. However, if you do end it you can go on getting Housing Benefit, although the Local Authority would need to be told what you are doing and you would need to let them know what you were living on. You should contact them as soon as you stop the ESA as otherwise they will be notified by DWP & will automatically stop your HB.
4. Your appeal will be dealt with by a Tribunal - not by an ESA advisor. It will be a good thing for you to attend the appeal hearing yourself to give them evidence and answer questions, in addition to making sure they have all the medical evidence from your GP, specialist etc.