Consider porting your sketch to esp8266 board like Wemos D1. The API of the ESP8266WiFi library is same as the API of the WiFi library.
To use esp8266 as network adapter, you have two wiring options.
If you have or buy an esp8266 module with SPI pins, you can connect the esp8266 over SPI like the WiFi shield is connected. Then you upload the WiFiSpiESP firmware to esp8266 and you include in your sketch WiFiSpi library instead of WiFi library.
Second option is to connect esp8266 over Serial or SoftwareSerial. The esp8266 should have AT firmware installed. The you can include WiFiEsp library in you sketch and initialize it with the Serial interface of your choice. You can find the WiFiEsp library in Library manager. Most new esp8266 modules have AT firmware installed at default.
There is one more option for Serial connection, if WiFiEsp wouldn't work for you. WiFiLink firmware and library
All mentioned libraries WiFiSpi, WiFiEsp and WiFiLink have the same API as the WiFi library so the rest of the code should work without changes.
The SPI connection takes a lot of pins, but it is the same with the WiFi shield. SPI connection with level conversion:
Comment to schema: SPI pins are on the ICSP header of UNO/Mega/Nano (digital pins 11, 12, 13). Pin 10 is needed as SPI SLAVESELECT and pin 7 is used only by SPI version of WiFi Link as SLAVEREADY signal from the esp8266 GPIO pin 5 (D1). For ATmega OTA upload GPIO pin 4 (D2) of esp8266 is used to reset the ATmega and Atmega pin 4 is connected to reset esp8266 from Atmega sketch.