If it’s a joint mortgage then surely there will only be one mortgage payment to the lender - so maybe the mortgage payment is coming out of a joint account that you both pay into?
If you remortgage then yes this will be for the entire mortgage amount not just his half or just your half, and the interest rate will be higher and so yes you may pay more in interest than you currently do each month; however, paying a big lump off the mortgage will reduce both the repayment part and the interest part.
For example, you might currently owe the bank £100k and currently have monthly mortgage payments of £800, and each contribute £400, with £300 being repayment and £100 being interest.
If H pays a lump sum off the mortgage, say £40k, leaving a mortgage of £60k, which at a higher interest rate might mean monthly mortgage payments of £560 a month, so £280 each, and that £280 might be made up of £155 repayment and £125 interest. So in that scenario you’d be paying more interest each month but less over the term of the mortgage due to the lump sum.
You need to run the exact figures and decide yourselves, what you want to do. Remember that paying a lump sum off will save you quite a lot of interest but will likely increase the monthly interest rate on what you have left to pay off.
Paying a lump sum off just before you remortgage is likely the best financial move rather than remortgaging earlier.