Um, not quite. Far as I've been told, the whole tattoo thing is some weird form of urban myth Jewish moms tell their kids so that they won't run out to get one. hillel article on it. There's also a secondary reason why people are allowed burial if they've got tattoos; if they do teshuvah (apologize to God/redeem themselves) for the act of getting the tattoo, it's all basically sorted out.
Prohibiting organ donations has as much to do with the belief in the messiah and the eventual resurrection of the dead, which works out better if everyone is intact, as with desecration of the dead. Because the messianic stuff is usually the reason for the most radical stances, most mainstream Orthodox Rabbis often don't have any problem with organ donation so long as it's going to save a life. A secondary reason for allowing donation but not advocating signing the card is because of a fear that death may be hastened if a person seems to be in a vegetable state and has an organ for harvesting. Hastening death is a big bad sin, 'cause God is the one in ultimate control of when time ends.
Just about every modern orthodox Jewish law traces back to the Shulchan Aruch