If you have enough money on hand, the frequency of being paid is irrelevant. I would recommend trying to have at least 3 months of expenses in the bank at all times.

You don't need to change your budget structure at all based on your payment frequency as long as you have enough money in the bank. The fact that your paychecks and bills will sync up less is not important so long as you have enough money.

If you move to a 2 week pay cycle, there will be a bit more variance month to month due to longer and shorter months. The "3 paycheck" month is literally irrelevant. It doesn't matter if you're paid on the 30th of one month or the first of another month. Your income is the same and the long term effect of being paid more frequently is that you get your money earlier. Some people treat the "3 paycheck" month like a bonus, but in reality, they are getting their first paycheck in the following month on the 12th or 13th instead of any earlier day, so treating it like a bonus is going to lead you to treat it unlike all of your other income.