Early warning. There are no public holidays in March so immigration offices will be open as usual Monday till Friday. However come April there are several holidays. Monday April 8th is a substitute date for Chakri day and offices will be closed.
Songkran holidays will be from Friday April 12th until Tuesday April 16th. Offices will be closed during those dates.
Depending on the office you use they will generally be manic prior and after these closure dates. If you can avoid using them during these times then it’s well advised to do so.
Land borders or airports are not affected by public holidays.
Plan accordingly.
TLDR : Answer Summary
In March, immigration offices in Thailand will operate normally with no public holidays. However, in April, they will be closed on Monday, April 8th for Chakri Day and from April 12th to 16th due to Songkran holidays. It's recommended to avoid using immigration services during the busy periods surrounding these closures, although land borders and airports will remain open.