The CRA's website has pretty good information on this type of thing. The search function is not great, however, so I recommend going to Google and typing:
Site:canada.ca [search term]
If you search
Site:canada.ca TFSA Transfer
It brings you here: https://www.canada.ca/en/revenue-agency/services/tax/individuals/topics/tax-free-savings-account/transfers/transfers-between-your-tfsas.html
"If you want to transfer funds from one TFSA to another or from one issuer to another, there will be no tax consequences if your issuer completes a direct transfer on your behalf. For more information, contact your issuer."
It seems that this is not something you are able to perform yourself. Unfortunately it seems you may need to go back to your issuer with this information and ask, again, that they perform the transfer on your behalf. Note that failure to comply with this request on their end likely has stiff penalties behind it, so it may help you to get the individual's name and wave that stick around to make something happen.
You may also have better luck by first opening an account with your new desired institution, and asking that they assist in requesting the transfer from your old institution. They have incentive to help you here as it's the only way they get to serve you, so you'll have valuable help on your side.