Hi Helen,
Worry ye not. I've noticed a similar finding in my recent runs wearing a HR monitor. I'm 34, and don't carry any excess weight, am reasonably fit for a novice runner. My waking HR is around 54bpm (has actually dropped from the high 60's a couple of months ago since I started running more often again) but when I do a slow run (the slowest I can run without walking) at the easiest possible effort, my HR increases to about 150ish within about 5-10 mins. I can sort of manage to keep it hovering at that rate, if i regularly check it, and keep reminding myself to slow down. it's harder than you'd think to run very slowly.
Not knowing what you goals are, I won't preach about what you should or shouldn't be doing, but I will say that if you were to steadily increase your mileage, such that you're running maybe 4-5 times per week, and if a couple of those runs were nearer 60 mins, then you'd almost certainly notice that in the space of a few weeks to a couple of months your HR at that same moderate level of exertion would drop. That's precisely where I find myself at the moment. I'm planning to run a marathon next April and want to spend from now until xmas running around 30 miles per week at a VERY slow pace with the aim of developing my aerobic fitness. You can find more on the theory behind this type of base training in the forums. search for "base training", "HADD", "endurance training" etc and you'll see lots about it.
Hope that helps.