I've used that web site to buy tickets to Europe two years ago, and also had computer trouble. I called LOT's New York Office (1-800-223-0593) and asked to be transferred to the web support. The guy who answered the phone was very helpful and was able to trace the problem and guided me through to complete the transaction. He also sent me a couple of mouse pads for my troubles.
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You may try call him and see if he can help. $350 RT to Europe is a good deal! AND you earn Qualiflyer miles!
BTW, they fly to Tallinn (if you know what I mean)!
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