I'm trying to transfer money to my BTC-E account from my bank (checking) account or debit card.
There's a reason bitcoins exchange for a lower rate on BTC-e than elsewhere, and that difference is partly because of the difficulty in transferring funds there.
AurumXChange will let you transfer VouchX USD vouchers as well as a few other redeemable codes (including Mt. Gox USD codes) into BTC-e vouchers.