Yes a stock will on average drop by the amount of a dividend. The actual change can fluctuate during the day just based on normal market fluctuations. But things like market stop and limit orders will adjust automatically by the exchange by the exact amount of the dividend due to this effect.
The reason that a stock drops after a dividend is because it has sent cash out the door. If a company is worth $100 per share, and gives $10 per share to its shareholders, that company is now worth $10 less (because of the cash out the door). From a shareholder wealth standpoint it's a wash since it now has a $90 share and $10 cash.
What the stock does from there is not guaranteed. It can go back up, or go down for completely unrelated reasons, but there's no fundamental reason that the stock would go one way or the other after the dividend adjustment.
(there may be some patterns that can be observed before and after dividend announcement/ex-div that can be exploited on average, but they are not guaranteed and there's no fundamental reason for a single stock to behave any differently just because a dividend has been paid)
I am looking to double my position in this stock but only if there's a ~7% correction on the ex-div date
From a wealth standpoint, it won't matter (other than rate and timing of taxes). Say the stock is at $100 today and has announced a 7% ($7) dividend, and you own 100 shares for a total value of $10,000. All else being equal, after the ex-div date the stock will drop to $93 and you'll be eligible for a $7 dividend. Your total wealth from this stock will still be $10,000 ($9,300 in stock and $700 in cash) less any taxes you'd owe on the dividend (but the capital gains you'd pay eventually on the stock would be lower as well). If you double before the ex-div, you'll still have $20,000 ($18,600 in stock and $1,400 in cash). If you double after, you'll also have $20,000 ($18,600 in stock, $700 in cash form the dividend and $700 "saved" from buying at $93 instead of $100).