I have never ever been flexible. I have never been able to touch my toes and I remember being utterly bemused in the 80s and 90s when exercise routines would involve that thing where you spread your legs out in a V, grab your toe and lower your head down to your knee. At my maximum flex I still look like I am sitting upright! Any Scots on here will recognise the description "stechy" 
For a while I was fairly cardio fit (ran a half marathon once) but I let that slide about 5 years ago and am just about to embark on a plan to improve my fitness again, but this time I'd really like to prioritise improving my flexibility as well as my cardio fitness. Generally I stick to running and swimming. Does anyone have any recommendations for a plan to make me flexible? It would need to be a total beginner approach as I am almost off the scale inflexible. Please don't say yoga, I tried it once and had real trouble with balancing and not having enough upper body strength. Just felt shaky and dizzy by the end.
Thanks!