Finally Debt Free

I've been bouncing around in debt for 5 years and finally managed to pay the majority of it off earlier this year. All that remains is £6000 on a low interest loan over 5 years which i cant pay in advance without penalty. This loan was for a car when my last one died a death and will be due up October 2020, its a manageable £225 a month.
Finally i can see the light at the other side and I am now looking at saving towards a house which was a complete pipe dream until recently, i moved from renting to living back with my parents last year which is what allowed me to free the cash up to pay off my debts but I am looking at a 3 year plan to move out.
I have £2500 in a low interest paying savings account that is tied to my current account but because of my financial situation have never looked at changing bank accounts etc until now. Ultimately everything is tied to the same bank (NatWest) and i have no idea how to go about getting a better deal.

I feel my best bet is to drip feed from a current account/savings account into a Help to Buy ISA. Can i do this by keeping my main current account and opening a high interest 2nd current account, transferring a chunk of my wage from one to the other and then finally into a HTB ISA or is this not allowed? Has anyone any other suggestions how to maximise my ability to save or should i just be looking at the highest paid savings account and swapping to that.

Thanks