Plenty of bikes about and I can agree that a more stretched out riding position does help some back problems. May not in your case so time to visit the shops- and get test riding. That would probably decide you on the style of bike to get and then you can really start searching.
And do not be put off by Race specification Frames. The lower more stretched out ride position of these may suit you.
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How long was I in the army? Five foot seven.
Spike Milligan