Firstly the currency of Hungary is the forint, not the fornit.
In these countries ATMs will give you the best exchange rate on the day that you withdraw, but if you don't have a card with no cash withdrawal fees it may be cheaper to exchange at home, or bring US cash, bearing in mind that any time Trump or Merkel or Macron or whoever says something the rates will change.
What I can say is that leftover korun or forint will not be a problem, as you can easily exchange them without losing out (at least in Prague and Budapest - YMMV if you leave those countries from some small town).
You can get spreads as low as 0.5% for both euros and US dollars. So if you exchanged $1000 then spent exactly half of your money, you might get $490-$495 instead of $500 if you exchanged the remainder back immediately. $5-$10 is not really worth worrying about, and you might even get back more than $500 if you are lucky.
Also the refusal of Americans to embrace the dollar coin works in your favor, as if you exchange to USD, you won't be left with more than $1 worth of those currencies.
You can try these two companies in Prague. There are plenty more similarly priced but do you want to waste time wandering from shop to shop comparing rates? Show the coupon on your phone to get the VIP rates.
https://www.exchange.cz/vip-coupon.php
Sm?nárna Nekázanka Exchange ? nejvýhodn?j?í kurzy valut a deviz.
For Hungary you can use
A legjobb valuta árfolyamok Budapesten és vidéken - Pénzváltók, pénzváltás, euró árfolyam,dollár árfolyam, valuta árfolyam ma, valutaváltó, pénzváltó helyek, valutaváltás, valuta, napi árfolyam, valuta középárfolyam to find the best rate (in Hungarian).
https://correctchange.hu/en/home/ is quite good and has several branches.
Without knowing your tastes, I guess an average spend might be 2500Kč per day in Prague and 25000Ft in Budapest