If you are definitely selling your own house then the remaining mortgage gets cleared automatically at that time and your new mortgage should be in joint names.
If you were in a position to keep your own house, you could possibly rent it out , remortgage it and release some of the 60% increase in equity. You could then use those funds as a deposit on the new house. How you'd structure that arrangement would probably need legal advice in case of any future despute with your partner. You still end up with the same mortgage amount but part of it is secured against your house and part against the jointly owned house plus you get to keep the original house as an investment. Of course there are tax implications re the interest and the rent but its possibly an option worth considering.