JS: displaying different timezones in specific format

Okay so I have a feature I’m working on for a new site where I need to display current time for both East Coast and West Coast. It needs to be display in the following way:
NY 11:24:21 PM
CA 08:24:21 PM

Thus far I’ve managed to get the time zones to start displaying but I need to make sure they both are pointing specifically only to EST & PST and that they are displaying inside the proper html blocks. I also need the AM/PM to wrap inside a html block.

Here is my sample code:

function showTime() {
    let date = new Date();
    let hours = date.getHours();
    let minutes = date.getMinutes();
    let seconds = date.getSeconds();
    let formatHours = convertFormat(hours)
    
    hours = checkTime(hours)
    hours = addZero(hours)
    minutes = addZero(minutes)
    seconds = addZero(seconds)
    let format = `NY ${hours}:${minutes}:${seconds} <small class="font-medium">${formatHours}</small>`
    let pstDate = date.toLocaleTimeString("en-US", {timeZone: "America/Los_Angeles"}, {timeStyle: 'short'})

    // Output Times
    $('.time-ny').html(format);
    $('.time-ca').html(pstDate);
}

function convertFormat(time) {
    let formmat = 'PM'
    if (time >= 12) {
        format = 'AM'
    }
    return formmat;
}

function checkTime(time) {
    if (time > 12) {
        time = time - 12;
    }
    if (time === 0) {
        time = 12;
    }
    return time
}

function addZero(time) {
    if (time < 10) {
        time = "0" + time;
    }
    return time
}
showTime()
setInterval(showTime, 1000)
body {
    min-height: 100vh;
    display: flex;
    align-items: center;
    justify-content: center;
    background-color: #000000;
    flex-direction: column;
}

.clock {
    font-family: sans-serif;
    font-size: 1rem;
    color: #fff;
}

.font-medium {
    font-weight: 500;
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div class="clock time-ny"></div>
<div class="clock time-ca"></div>

Any help from you real JS coders would be super appreciated. I dabble in this stuff but it will never be my bread and butter.