What Google is telling you, and what to do about it
The message on screen narrows the problem down quickly.
Google cannot find the record
By far the most common case. Run the lookup: if the record is missing, it is in the wrong place or the host field changed it; if it is there, Google needs more time. Either way, adding the same record again in more places makes things harder to untangle later, so fix it in one place.
Google says the domain is already in use
This is a different problem. The domain is already attached to another Google account, often an old trial someone started years ago, or a Google account set up by a past employee or web person. Google has a recovery process for this, and it also starts with proving you own the domain through DNS, so the same nameserver lookup is still the first step.
It verified, but email still arrives at the old place
That is expected. Verification does not move email; the MX records do. Until they change, mail keeps going to your current provider, which is what you want while the new mailboxes are being set up.
Google offers another method
Google also lists other ways to verify, such as a CNAME record. They all depend on the same thing: changing records at the provider that answers your DNS. If the TXT record failed because it went to the wrong place, another method will fail for the same reason.