Hi Hester,
Most likely you're mixing strings with dates, which won't compute nicely. Date objects are pretty strict, it's usually best to approach conversions by adding/subtracting a separate date object and then converting to a string:
const [timeZone] = new Date().toString().match(/([A-Z]+[+-][0-9]+)/);
var offset = new Date().getTimezoneOffset();
const date = new Date('August 29, 2022 16:00:00'); //our date to be converted


var offsetDate = new Date(date*1 + offset*60*1000)  //creating a new date by converting our original date and offset to miliseconds and adding them together
var offsetDateText = offsetDate.toLocaleDateString() + " " + timeZone //a string of the new date

console.log(offsetDateText) // "8/29/2022 GMT+1000"
Tinker with options as need be.
Good luck!

bgooldfed thank you so much! The formula to convert to a local date was especially where I felt utterly lost.

Below are some changes I made for posterity and in case it's beneficial for others.
const [timeZone] = new Date().toString().match(/([A-Z]+[+-][0-9]+)/);
var offset = new Date().getTimezoneOffset();
const date1 = new Date('August 29, 2022 14:00:00'); //our date to be converted
const options = { weekday: 'short', year: '2-digit', month: 'short', day: 'numeric' }; // some additional formatting to show weekday more clearly

console.log(timeZone) // GMT+1000 Logged and embedded timeZone seperately, also because it gave me more control over formatting in the survey
Qualtrics.SurveyEngine.setEmbeddedData('timeZone', timeZone);

var offsetDate1 = new Date(date1*1 - offset60*1000)
// creating a new date by converting our original date
// and offset to miliseconds and adding them together
//Adjusted the calculation to state - offset*60*1000 instead of + offset*60*1000 I think it's because offset returned is negative when +GMT and positive when -GMT. This seems to work ‍♀
var offsetTimeText1 = offsetDate1.toLocaleTimeString(e], {hour:'2-digit', minute: '2-digit'}) /*+ " " + timeZone //a string of the new date*/

// Added toLocaleTimeString so that I could have more control over formatting for date and time. Also commented out timezone as I'm embedding and adding a field separately for formatting

// Added undefined locale and date formatting options
var offsetDateText1 = offsetDate1.toLocaleDateString(undefined, options); //a string of the new date

console.log(offsetTimeText1)  // "14:00:00"
console.log(offsetDateText1)  // "Mon, 29 Aug 22"

Qualtrics.SurveyEngine.setEmbeddedData('offsetTimeText1', offsetTimeText1);
Qualtrics.SurveyEngine.setEmbeddedData('offsetDateText1', offsetDateText1);

For anybody else visiting this thread struggling with dates, bear in mind that they are just a number expressed in miliseconds. So if all else fails, working in miliseconds should be reliable.