The first sentence, "She was just putting on a cockney accent for effect, though she wasn't," suggests that she's using the accent to emphasize something or accentuate her point. 'For effect' here means 'done to add emphasis'. In the other two, "She was just pretending a cockney accent for drawing attention, though she wasn't," and "She was just pretending a cockney accent for attracting attention, though she wasn't," the meaning comes across that she is using the accent not to emphasize her point, but to make or encourage people to look at her and pay attention to her. 'To draw attention' and 'to attract attention' both mean to encourage people to pay attention to you. One other note I'd like to make is in American English we normally don't use 'pretending' with a noun, we follow it with an infinitive. You could also use 'imitating'. Additionally we don't say 'for drawing attention' or 'for attracting attention' - 'to draw attention' and 'to attract attention' sound better. So that would be: She was pretending to have a Cockney accent to attract attention, though she wasn't. She was pretending to have a Cockney accent to draw attention, though she wasn't. or She was imitating a Cockney accent to attract/draw attention, though she wasn't.

'for effect' means that she was trying to put a cockney accent to build onto a result to something. 'drawing attention" is basically same as 'attracting attention', but this is a more proper way to say this sentence: She was putting on a Cockney accent to attract attention, but it she didn't.