My DM is telling me a natural 1 is worst possible result, but when my build reaches Sorcerer 8/Bard 1, my Persuasion bonus will be +10, meaning my worst roll should be an 11. Right?
... and so is your DM, technically. A 1 is the lowest you can roll on the d20 die: it is the worst possible result for that roll. But your modifiers can affect that greatly.
If your character gets a +10 by a certain level, at that point he will auto-pass all DC 11 or lower skill checks that use your skill. This is because critical failures for ability checks do not apply. Those are reserved for attack rolls and death saves.
RAW, there are no critical success/failures on skill checks or saving throws, so while rolling a nat-1 would be the worst possible outcome, with a +10 modifier is would still beat out the average roll of an unslilled commoner (+0).
However, many tables play with a house rule, where skill checks do have crit success/failure. If that's the case, then yes, your DM is correct: rolling a nat-1 will cause you to fail, regardless of your skill modifier, just like a nat-20 will cause you to succeed regardless of your skill modifier.
Using his words, if I tried to persuade a dragon(which he knows I'm planning to try to befriend a wyrmling red dragon) if I rolled a nat1 then it mean's I've said the worst possible thing I could have as a critical fail regardless of the bonus.
You should clarify this with your DM. It is either a misread rule, or a house rule.
Your DM might use a house rule that a natural 1 is a critical fail for an ability check. By the rules as written, is is not - the "critical fail" and "critical hit" rules are applied to attack rolls and death saving throws, but not to ability checks.
If your DM interprets this rule incorrectly, they can apply it to ability checks as well.
Rolling a 1 is the worst possible result. For persuasion, you have said the worst thing you could have said in order to convince the other party. However, since you have a +10 to the roll, the absolute worst thing you could have said would still be convincing if the DC is 11.
Basically, you're so talented in speaking, even if you mess up what you're trying to say, you're still pretty convincing.