Trantor,
Assuming that your lines were relatively new, you should not have chafed four lines in a short period of time. As you know, chafing is almost always the mode of line failure and something appears to be accelerating this in your case. While double braid may chafe a little faster than 3 strand, either should work well in normal docking, even in rough harbors. It seems you may need the extra shock absorbing of the more stretchy 3 strand over the characteristics of double braid, but if you went through four lines in short order on double braid, you're likely to have some problems with 3 strand. Examine exactly where the lines failed. If there is a rough spot or sharp edge, even a very small one, it will cut through a line that is flexing in a short time. A smooth round head stanchion bolt did the job for me once. Are you using proper chafing gear at the contact points?...not plastic hose. How are the other boats of comparable size and exposure to the waves in your marina handling the situation. IF something is working for them, it'll probably work for you. Did the loose cleat bolt figure in on the chafing by contacting the line...why did it become loose? Any boat supply store will know what kind of line that they are selling, so you can go there and see what the manufacturer has marked on the reel of line or on the boxes for the line. Good luck.