If you have an excuse that keeps you from fasting the six days of Shawwal in Shawwal because you have to make up missed fasts, then you may fast six days of Shawwal in Dhul-Qa'dah

There is nothing wrong with making up a missed Ramadan fast in the last half of Sha’ban.

It is permissible for you to make up missed Ramadan fasts on the day of doubt and one or two days before Ramadan. For more, please see the detailed answer.

If you start an obligatory fast, such as making up a missed Ramadan fast or offering expiation for breaking a vow, it is not permissible for you to break the fast without a valid excuse, such as sickness or travel.

It is not essential to make up for missed fasts consecutively. If he does that, it is better, but if he does not there is nothing wrong with that.

The majority of scholars said that it is not permissible to fast on the Days of Tashriq whether it is an optional fast, making up a missed fast or fasting in fulfillment of a vow. So, the fasts that you observed on the Days of Tashriq to make up for missed Ramadan fasts, are not valid, and you must repeat them.

If you did not fast three days of Ramadan out of heedlessness and not because you believed it was permissible to do so, then you have committed a serious major sin by violating the sanctity of Ramadan and you have to make up for these days later.