I didn't realise there was a difference between amniotic fluid and waters? In fact I'm sure there isn't..
I've been having the same thing for several weeks, I'm 36+6. I've had a growth scan which showed the baby was fine but the water was low.
I'm having another on Tues. If the water is any lower they will want to talk about induction, though I don't think at 'term' this improves outcomes anyway, so will try and avoid it unless things are truly desperate.
The Dr I was seen by also said, no it's just probably discharge, but it's continuing and getting heavier...just not enough to make me think it's forewaters, which is the big amount - it's got to be hindwaters, that's the small area of fluid (there are two different bits apparently) and tends to cause a trickle rather than a big ol' gush 
Don't panic at this point (realise you posted on Thurs, hope you are sorted now anyway) as there is a very low risk of infection with hindwater leaks and they can often reseal and replenish without any problem.
But do go and get checked out, especially if you manage to fill a pad, or it gets any heavier, or you start having contractions etc.
At 37 weeks your baby will be ready to arrive anyway so that's good.