There are a few unusual aspects of your problem.
You and your doctor are correct that if itching is the dominant symptom, it is likely that allergy is at the root of the problem. On the other hand, allergic problems tend to affect both eyes and not just one. Typical allergic eyes may have a watery discharge but not usually thick or stringy. Allergic problems may be associated with blockage of the tear drainage pathway and therefore overflow tearing.
Are we absolutely certain that this is an allergic problem? Is there something that could be getting into the left eye only (left hand, sleeping on your left side) and not the other. Or could this be an eye infection or some other process with an itching component?
It is much better to find an answer than to use steroid eye drops for an extended period of time. And of course, while using steroid eye drops, you need to be closely monitored by an ophthalmologist. The drops can cause your eye pressure to go up, can prevent your body from clearing an infection and over a very long term can cause cataract.
If the condition persists I suggest a consultation with an expert. You can find one by searching on www.AAO.org by selecting "Find an Eye MD" tab on top and look for "corneal/external disease" and your location to find an ophthalmologist with special training in this particular area.