Disregard my reply below, it has multiple repetitions and I don't know how that happened… Here is what I wanted to say:
I am having a similar problem – but not running or headphones –I, like others, am having a charging port / cables problem -and now a charging CASE problem – I do keep my phone in pocket & like you, am active: I have very busy, dusty, Outdoor work. Probably as result of latest iOS revision, one by one, charging cords, some EXPENSIVE, (6 foot, cloth covered, & also, a very buff brightly colored ribbon cable, & also "cheapies" alike) were said to be "not certified".
I HAD, it should be noted, previously used a totally waterproof Otter case for prior six months, on my first 5S, – until some good pictures I took came out soft focus, and hated flare issues from flash with that case (soft focus, because of case's perennially dirty lense window) (though NO charging port issues with Otter Metro, non waterproof, case, on my first 5S).
I quit the waterproof case some months ago, when some good pix I took came out soft focus, because of case's perennially dirty lens cover. Soon after, port issues surfaced for first time.. Flipping cable insertion direction helped, for awhile... Then, when i was ready to return the 6 or 8 mo. old 5s, because NO cord would charge, but one, and not even that one, w/o carefully weighting cord, to put vertical pressure on the still-functioning pins, on that last working cable.
At the T-Mobile store, found 100.00 Mophie charging case worked perfectly with the (damaged?) phone, as I waited for replacement to arrive; keeping my phone charged all day w/o having to nurse its tender charging port. - but I returned the phone on insurance...due to port issues with other cables... Now, brand new (Referb, mailed to me by TMobile's ins. plan) 5s is now giving the same "unsupported" message with cankles AND BRAND NEW Mophie case - I returned the Mophie case for brand new unopened one; due to its (unrelated, I think) failure; when18 hours later – I am having trouble, and the unsupported message! No abuse of either new phone or case, by me. – and I AM FRUSTRATED ALSO!
It seems that this "charging port, unsupported thing" is costing Apple, and every company downstream from Apple, plus users, lots of money: people who just want to buy - or produce - some cool accessories not thought of or available from Apple...For example, - this Mosie case, was set to make my phone finally run all day on a charge… Since it does not normally do so, when I give it to good use! -Which includes: taking pictures of my work, posting them to Twitter, and using GPS driving directions, and Expensify for mileage logging & receopts…I additionally hoped the Mophie charging case, with its micro USB charging input, would help me no longer have to constantly plug in my IPhone throughout the day, which can only increase damage to a (badly designed?) new charging port /wire system? - never had 30 pin problems with my 3G!...)