Coughing has various causes and can be elusive to diagnose. One simple cause that is often overlooked is that there really is something down there irritating your tissues, something that can't be dislodged easily. Sure, this might be a sign that your tissues are too sensitive due to some other cause that hasn't been diagnosed, but it's no reason not to get the damn thing out, whatever it is. My throat is sensitive due to acid reflux (GERD) and when it gets irritated I gargle with plain hot tap water a few times and that often gets rid of whatever was bothering me.
Another cause is, obviously, GERD or acid reflux. You might alter your diet slightly in the winter (most of us do after all) or your metabolism might change due to the weather, so you don't have the problem during the other seasons. Be sure to have an otolaryngologist or a gastroenterologist check for that. It can be checked with a couple of different kinds of scopes but your family practitioner doesn't have either one.
As for the possibility of asthma, my wife and I both have it; mine started in my late 40s and she was almost 60 when hers set in. So the fact that you've never been diagnosed before is no guarantee that you don't have it now.
If that doctor cured it with antibiotics, I assume you went back to the same guy when it came back. What did he say then?
If your prescription truly is "try to stop coughing," then all you need is symptomatic relief. Cough drops work pretty well. If not, dextromethorphan works pretty well for three or four hours. In the USA that's the standard non-prescription cough syrup.
If you need something stronger, for the daytime you can get it mixed with phenylephrine, which I believe is a vasoconstrictor that dulls your pain nerves. It also acts as a stimulant in many people (like me) so be careful. That combination is sold over-the-counter as DayQuil in the USA.
For nighttime, they sell NyQuil, which is dextromethorphan mixed with doxylamine. That's a sedative that will help you not wake up when your tissues are itching. I find it to be a very strong sedative; it's supposed to only be good for six hours but I still feel foggy twelve hours later. Again, be careful. Unlike caffeine, alcohol and nicotine, other drugs can't be counted on to have more-or-less the same effect on everybody.
Both of these mixtures also include acetominophen, generic Tylenol, which is an all-around painkiller and a mild anti-inflammatory, both of which are sensible ways to medicate an itchy throat.
Having an ailment that only needs to be treated symptomatically can be annoying because you'd really like to just get to the root cause and make it go away. But considering the variety of root causes out there today, sometimes maybe you're lucky that all you have to worry about is getting a cheap non-prescription analgesic at the drugstore.
