I'll just give some examples. The Prime Minister was really offhand with her junior ministers. She dismissed them casually, with hardly a thought. (it's usually quite rude to be offhand with someone. You are too casual, you don't take them seriously, you are abrupt and dismissive) Gee shucks! We've lost the game again. (Similar to "Aw shucks! It's an expletive, but a mild one. Gee! and Aw! are just noises we make. Shucks! is often said on its own. In fact, "Gee" and Aw! are often said on their own too. Exclamations of surprise or disappointment). An "offhand gee shucks kind of way" would be casual, and in this case perhaps a bit embarrassed. Sounds as if the person is trying to be apologetic, or not sound too arrogant. Actually, it would help to know what the CHARACTER of the person is like, and the situation. You could mean a number of things by, "Gee shucks....

Offhand means without thinking about it first. The expression "gee shucks" is an explicative that replaces curse words to make it less offensive. This part of the sentence means the same as saying, "I mean that seriously."