You do need a Turkish visa because you are changing airports and will not remain airside. Check-in staff was right.
If your ticket had your transit on one single airport you wouldn't need one but with the current itinerary you do need it.
I don't know what to do
Solution: Get your agent to change the ticket so your connection is on a single airport, for example Ataturk International. It will cost more but it will be easier than getting a transit visa quickly.
Transit - Turkey (TR)
Visa
Visa required.
TWOV (Transit Without Visa):
Passengers with a confirmed onward ticket for a flight to a third country within 24 hours. They must stay in the international transit area of the airport and have documents required for the next destination.