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First, rule out any fault with the HDMI cable by using a different HDMI cable or a new cable.
If that doesn't help, test the Xbox on a different TV to check that it gives you a good resolution elsewhere.
If it works ok on a different TV then you will know that the problem likely lies with how your TV is processing the input signal, and you could try using a different input cable type (e.g. Component cables)
If it is only happening with the Xbox and only on this TV, then it does seem likely that it is a HDMI handshake problem.
This happens when the TV gets confused and cannot process the signal correctly.
Try leaving the TV unplugged overnight and also remove the HDMI cable from the rear of the TV.
Turn on the TV the next morning, then connect the HDMI cable to a different port than previously used.
Let me know if that resolves the issue...
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