I solicit correction from kind fellow posters who speak Spanish but I am under the impression that Cereth thinks shoganai and dasai are synonymous. They are not.
Dasai is tacky, dweeby or countrified, i.e., the opposit of cool, hip, groovy or chilled. Very simple.
Yet, shoganai needs more detailed explanation. Ellipted from "shiyoo-ga nai" (sorry guys, I cannot wield my favourite macron in this PC), it literally means "there is no way." But no way to do what? This question leads to the many usages of the expression.
Shoganai is "I cannot help it," when used to describe an irresistible urge one is feeling.
チョコレートが食べたくてしょうがない。
chokoreeto-ga tabetakute shooganai.
I am craving to eat chocolate.
It can mean, "There is no use in doing something."
いまから書いてもしょうがない。締め切りは昨日だった。
ima-kara kaitemo shooganai. shimekiri-wa kinoo datta.
There is no use in [start to] writing from now. The deadline was yesterday.
Another meaning is incorrigible or impossible after the sense of "no way to mend."
あの人はしょうがない。
anohito-wa shooganai.
He is impossible.
[note: on a second thought, this use of shooganai as expressing something is disagreeable is not far from dasai.]
What seems to be derived from this sense is, "really."
夏は蚊がうるさくてしょうがない。
natsu-wa ka-ga urusakute shooganai.
In summer mosquitos are really annoying.
Finally, if uttered by itself, shooganai is an expression of resignation, "I cannot complain."