'tsui',according to me is used when you get indulged in your inner feelings so much that you forget what is going around. ex-夢中で、つい大きい声で歌ってしまって、はずかしかった。
Yes. The つい means すぐ。This emphases something happens very close in time/distance. つい２、３時間前に as recently as a few hours ついそこまで休憩に行ってきました。I went right over there for break.
I always thought that was ついに？ Am I wrong or can you drop the に？
ついに is different from つい. ついに can be replaced with やっと at last ついに買っちゃった！ Finally, I got it!
「つい」and「ついに」is different. 「つい」means... "regretfully" for example, you go shopping, you find a new PC. you already have a PC, but you buy the spanky new one anyway. then, you say 「つい、買った。」 in fact, you don't need that PC. 「ついに」means..."at last" you wanted a PC for a long time, and eventually you get one. then, you say, 「ついに買った。」 you really need that PC. :( tsui = ukkari :) tsuini = yatto