I will tell about Nikolaev, though it's pretty similar in all the cities.
Never exchange more money than you need to pay for your current expenses at the airports and stations. This is the worst exchange rate ever. In Nikolaev you won't find good rates in the banks either, look for small exchange offices in the central streets, they are kind of legal and they show more reasonable rates. And the best rates are at the Central Market where we have black market for the currency. Though it seems illegal and sometimes dangerous.
Here is the difference in rates when you buy hryvnia for dollars:
Banks - 16.00
Exchange offices - 19.00
Black market - 20.00
Rodan is right. I lived out of cash machines in Nikolaev and Odessa summer, 2014. My American dollars I brought from home were not acceptable, as they were not new enough. The exchange rate last summer was 13.5 to 1. How things have changed...it was 4.7 to 1 in my first visit to Ukraine in 2007
I remember 2007 rates but then again, a litre of petrol was less than 4UAH. despite the rate change, the price has remained very stable in Ukraine, not like US
although there are many places selling at 15.99 for Euro95 grade reflecting lower global gas prices but the difference between US and UA is that gas stations make a good profit on gasoline.
Rodan is right about exchange. your American bank WILL screw you. exchange your highly coveted US Dollars locally