many schools collect CVs over a long period of time and only reply to applicants much later in the year - is this true?
Yes, that's true. Most jobs in schools start in October and last until June. There is precious little work to be had during the summer, and what little there is has already been snatched up by more established teachers. Furthermore, many schools pay reduced wages for summer courses. So don't waste your time trying to find work for the summer.
And, if not, at what point should I give up and look elsewhere?
The boat for English teaching in Poland sailed long, long ago. The economic crisis has shrunk demand significantly while simultaneously increasing supply as unemployed British and Irish slackers, drop-outs and recent grads have gravitated to the larger, attractive cities in Poland in search of cheap beer, easy poontang, and jobs. They have driven prices down quite a bit as they are willing to give lessons for 20 or 30 PLN an hour out of desperation. Work in schools has also decreased as fewer companies are generous in footing the bill for classes for their employees. Also, there are more native Poles that can do a decent job of teaching English at the lower levels, so schools are less willing to pay extra for native speakers.
Forget about the large cities especially Warsaw, Kraków and Wrocław. The competition is to fierce there for a 23-year-old to become established whether as a teacher in a private school or as an independent tutor, so forget about that path. You might get a job in a lousy school for lousy wages, thinking that it would at least be a foot in the door. Sorry, but working for a crumby Callan method, Direct method, Avalon or Berlitz school just ain't gonna open up any doors.
If there's still opportunity to be had for native speakers, it's in small towns off the beaten track way out deep in the provinces, in places that are never mentioned in the tour guides, especially in Eastern Poland, where native speakers are rarely seen, and therefore are still in demand. Places like £omża, Chełm, Sejny or Limanowa. Of course, these jobs are a lot harder to find and are riskier, but the pay to cost of living ratio is higher than in the bigger cities.
My advice would be to give up English teaching altogether and go back to school to earn some salable qualifications while you are still young. English teaching is going to do little for you in terms of career development, and as a career, it has no future, even in places like the Far East. If you're doing it for a year or two for vacation, adventure or just plain ***************, have fun. But don't expect to make any money from it, and you won't be disappointed.