You could and should shop around, using moneysupermarket or another price comparison site. To do this effectively and make an informed decision, you need to use your actual bills (eon may have sent you their projection of your annual use over the next year, which may help, if it is accurate). Whatever you do, don't use the option of inputting how much you pay monthly. This is your regular payment, and may not reflect your usage. Once you have your quotes from various suppliers factor in the cost of the switching penalty. You may not even have to pay this if you have just been informed of a price rise, which gives you a get out of jail free card, if you switch within the statutory period shown.
Many customers will feel some sympathy with people on JSA stuck with bills that they will struggle to pay, but, as far as I know, there are no 'social tariffs' for people in this position (unless someone knows different). Only practical advice is reduce usage, use low wattage eco-friendly light bulbs, very cheap, a good idea as they should pay for themselves in time anyway. You could also remove a couple of bulbs from multi bulb light fittings. Turn off all lights not in use. Turn the heating down a couple of degrees, and wear an extra layer of clothing. This is what many pensioners (no, I am not one!), do regularly. Also investigate on web subsidised house insulation, which can attract government grants. All the solutions are in your hands, and should significantly reduce your gas elec costs.
All of this is good housekeeping advice for all, not just those feeling the pinch. Good luck with the job hunt.