Bad path not working how I think it should in C++ function

I am taking a C++ programming course and working on a project. I am having an issue with the bad path of this problem. It is a user based input problem for temperature conversion. I have the happy path working great but the bad path is not acting like I expect it to. To my understanding, if I hit the bad path, it should wait for me to re-enter another input before exiting. In my code though, it just prints all the info in the bad path and exits.

#include "unitConversionHeader.h"
#include <iostream>
#include <iomanip>
#include <cmath>
#include <typeinfo>
#include <any>
using namespace std;

void tempOutputs(int, char);

void tempConversion() {
    // Parent function.Collect the inputs, handle the conversions and checks then return the result.
    int temperature;
    char units;
    char retry;
    char killswitch;
    bool running;

    running = true;

    cout << "This program converts Temperatures from Fahrenheit to Celsius and vice versa. \n";

    while (running) {
        cout << "Please enter your temperature: \n";
        cin >> temperature;

        cout << "\n";
        if (cin.fail()) {

            cout << "The value you have entered is incorrect or not supported. Would you like to try again? (Y/N): \n";
            cin >> retry;
            units = toupper(retry);

            if (units == 'Y') {
                break;

            }
            else {
                cout << "Too many incorrect attempts. Exiting program now. \n";
                cout << "Thank you. Goodbye. \n";
                running = false;



            }
            
        }

        else if (cin.good()) {
                cout << "Please enter your units (F/C): ";
                // convert to upper to eliminate risk of inputting the wrong value, lowercase is sufficient
                cin >> units;
                units = toupper(units);
                cout << "\n";

                if (units == 'F' || units == 'C') {
                    tempOutputs(temperature, units);
                    cout << "\n";
                    cout << " Do you want another conversion? (Y/N): \n";
                    cin >> killswitch;
                    killswitch = toupper(killswitch);
                }
                if (killswitch == 'N') {
                    cout << "Thank you. Goodbye. \n";
                    running = false;
                }

            }
        }


    }




    
void tempOutputs(int temperature, char units) {
    // After receiving the inputs we add the conversion logic and then run the calculations
    string lhs = "Celsius";
    string rhs = "Celsius";
    int calculation;

    if (units == 'F') {
        lhs = "Fahrenheit";

    }
    else {
        rhs = "Fahrenheit";
    }
    // taken from header
    calculation = unitConversion(temperature, units);

    cout << "A temperature of " << temperature << " degrees " << lhs << " is equivalent to " << calculation << " degrees " << rhs << ". \n";
}

When I run the problem, if I enter a char for the first input and hit the cin.fail logic I just get the following output:

This program converts Temperatures from Fahrenheit to Celsius and vice versa. 
Please enter your temperature: 
f

The value you have entered is incorrect or not supported. Would you like to try again? (Y/N): 
Too many incorrect attempts. Exiting program now. 
Thank you. Goodbye. 

What should happen is that it hangs and waits for me to enter another value to confirm how I would like to proceed.

Before these statements

        cout << "The value you have entered is incorrect or not supported. Would you like to try again? (Y/N): \n";
        cin >> retry;
        units = toupper(retry);

you should write

#include <limits>

//...

std::cin.clear();
std::cin.ignore( std::numeric_limits<std::streamsize>::max(), '\n' ); 

to clear the input buffer. Otherwise the variable retry reads the current content of the buffer that according to the provided by you output is the letter f..

Also it seems using the break statement

        cout << "The value you have entered is incorrect or not supported. Would you like to try again? (Y/N): \n";
        cin >> retry;
        units = toupper(retry);

        if (units == 'Y') {
            break;

contradicts the prompt.