over drawn bank statements would probably greaty increase your risks of refusal. thou showing you have savings may reduce them risks.
as for transfering money from one account to another , beware, If there are disproportionate deposits and withdrawals in the bank statements, they should be explained in the sponsor's letter.
you might not like what i'm going to suggest, but i'd transfer enough so you've got at least 2 or £3,000 and wait 6 months so you've got 6 bank statements with at least £2,000+ ..
or you could explain you have savings but i think the risk would be greater than if you waited..
