As explained in Is a Schengen visa/transit visa required for a transit flight through two Schengen airports? you will not be able to stay in the international area of the airport. As you will go through a passport check and enter the Schengen area, an “airport transit visa” (type A) is definitely not enough and you need to apply for a regular uniform Schengen visa. On the form cross “transit” as the purpose of your trip (and not “airport transit”).
Unfortunately, if the German consulate covering your area is not able to offer an appointment or issue a visa in time, there is not much you can do. You can always try to contact someone at the consulate to see if there is still some way to set up an appointment but it seems unlikely they would make an exception.
You could also try to contact the airline and hope they refund the ticket anyway but they most likely don't have to (conditions of carriage typically specify that securing all required visas is the passenger's responsibility).
Alternatively, you could try to change your connection to have only one stop in the Schengen area (i.e. find direct flights from Chicago to Frankfurt and from Frankfurt to New Delhi or even to another US or Indian destination). It will probably cost you more and you will have to pay some change fees on top of the ticket price but you might be able to save your trip and avoid losing all the money you already paid. Since you would still be flying with Lufthansa, they might be more willing to accommodate you.
If you have only one stop at a German airport and direct flights out of the Schengen area, you won't need a visa at all if you have a valid US visa.