You are correct.
What DOES transfer into the new common position if you exercise is the COST BASIS from the warrant. In other words, if you pay $6 for a warrant with exercise price $5, and then exercise it, you cost basis is then $11 in the common (and, as you correctly note, the basis date for that common is the date on which you exercised the warrant).
If I could afford to buy all of them, I would not need to buy any of them and I sure wouldn't be spending time on the message boards!