Sufferers of balance disorders should qualify for The Disabled Persons Railcard – this is for people with a disability that makes travelling by train difficult.
The card costs £20 for a year or £54 for three years. The Railcard will allow both you and an adult companion to one third off most rail fares throughout the UK.
See https://www.disabledpersons-railcard.co.uk/ for more details. The application process may require some level of determination to ensure that hidden disabilities are recognised. Do let the support group know if you encounter any difficulty with this.