They will apply any payments to the highest rate debt first - this is the rule, so it should work, as long as there aren't any T&Cs to prevent this.
It's also a good idea to transfer your overdraft over in this way, providing that you don't run it back up. It's better to have credit card debt than overdraft debt in the eyes of the lenders, and you've probably seen the news about overdraft rates increasing massively soon, therefore everyone should change their thinking about overdrafts and credit cards, and aim to completely avoid overdrafts.
However, I've applied for a couple of things recently and don't want it to negatively affect my credit rating
Can I ask what you're saving your credit rating for? At the moment, you're in debt so your number 1 priority should be to reduce the cost of that debt, and it's a good use of risking/investing your credit rating in that by looking for a cheaper balance transfer deal.
However, I'd be cautious about your plan to pay off debt when you sell your house. That is indeed a good plan, but you do need to bear in mind that you've either transferred unsecured debt to secured debt, by increasing the amount of the mortgage on your new property, or if you're not taking out another mortgage, you've effectively used your equity as a 'get out of jail free' card, which may be an issue if you haven't addressed the reason that you got into debt in the first place if this was an overspending issue rather than a one off loss of income/high childcare costs that have now passed or something else that no longer applies?
Definitely aim to stay out of debt, and build up a savings buffer going forwards, because if you get into debt again, you'll have debts and a mortgage that is bigger than it should be, leading to bigger potential problems. 