Using async in googlescripts webapp

I am building a small web service version inside google scripts

I have 2 questions I couldn’t find an answer to,

1. Logging, I have tried Logger.log and Console.log
   But whenever I am calling the script using API, it runs but not logging
   And the execution record is not expandable to see any logs.

How can it be logged? in order to debug and so.

2. I have noticed that I can’t use ‘async’ functions,
   (from some other posts and by getting errors myself)
   Now, I need some of the requests to use fetch, which will require await,
   How can it be approached?

Here’s a code of mine using a simple post-request
which is meant to shorten a URL

    function doPost(e) {
    logSpreadsheet(e)
    Logger.log("doPost initiated");

    let reqbody = e.postData.contents
    reqbody = reqbody ? JSON.parse(reqbody): reqbody;
    
    let action = reqbody.action;

    if (action === "shorten_url") {
        Logger.log("shorten_url function initiated")
        if (reqbody.url) {
            let shortened_url = shorten_url(reqbody.url)
            return ContentService.createTextOutput(
                JSON.stringify({"shortened_url": shortened_url}))
        }
        else {
            return ContentService.createTextOutput(
                JSON.stringify({"error": "`url` parameter is missing"}))
        }
    }
    else {
        return ContentService.createTextOutput(JSON.stringify(
            {"error": "`action` parameter is missing"}))
    }

}

function logSpreadsheet (e) {
    let ss = SpreadsheetApp.getActiveSpreadsheet();
    let sheet = ss.getSheetByName("Logs")
    sheet.getRange(sheet.getLastRow()+1,1).setValue("test")
}


const shorten_url = async (source_url) => {
    console.log("shortening_url")

    encoded_url = source_url ? encodeURIComponent(source_url) : "";

    let res;
    let shortened_url;
    let cuttly_apikey = '***'
    let url = `https://cutt.ly/api/api.php`
    let params = `?key=${cuttly_apikey}&short=${encoded_url}`
    let options = {
        method: "get",
        mode: "no-cors",
        headers: { 'Content-Type': 'application/json' }
    }
    try {
        response = await UrlFetchApp.fetch(url + params, options)
        let resData = response.getContentText();
        if (resData) {
            response = JSON.parse(response);
            console.log('cuttly response:', response)
            shortened_url = response.url.shortLink
        }
    }
    catch (err) {
        console.log(`Error getting user data!n${err} `);
    }

    return shortened_url ? shortened_url : source_url;
}

Thank you for helping.