As we sell our current home and realize some capital appreciation, we want to withhold some money from our purchase so we can do some updating/addition to the new home. How does the calculation of tax liability work on this? Thanks!

You may do that aslongas you sell your primary residence, you can make up to $250K($500K for MFJ)in profit if you're a single owner, twice that if you're married, and not owe any capital gains taxes to the iRS/state; even better, there's no limit on the number of times you can use the home-sale exemption. In most cases, you can make tax-free profits of $250K (or $500K depending on your filing status) every time you sell a home.
Note;s you can see, your home improvements/addition can lower your capital gains by increasing your home adjusted basis when you dispose of it later.