Andy, I've also been on JSA in the last year, so really think you're on the right lines of 'refreshing' yourself. I remember how drastic it felt... A trip away on your bike will give you more than you can imagine!
Crankarm has the right idea about woofing and such.
Bikes don't have to be expensive to be reliable. I picked up a fairly good steel bike from a guy on Gumtree for £40. See what you can find, with a wee bit of searching.
Getting out on my bike was true escapism from grey days searching for a job. If the woofing thing doesn't appeal, just go away for as long as you can here in UK. Some gorgeous riding to be done here, not just in France! I did a 19 day tour here in Britain last year, which cost me a total of £400. I need to stress that I didn't wild camp until Scotland, and *did not* watch the pennies when it came to food. My point is, I could have easily restricted myself and spent between a third and a half of what I did.
I plan to go away for 5-7 days in September and not spend a penny (as I haven't got any to spend [as I'm saving for a new road bike!]) - we'll have to see how that turns out. But my point is, cycle touring is amazing because it's something you can do on a low budget.
Good luck either way.