My old Panasonic answering machine had a record-call feature to it, and it put out the beep tone every 15 seconds so your taping is legal. This machine used tape cassettes and I don't think they use them anymore. A better grade electronics store (not Radio Shack) might have such equipment.
Radio Shack sells a suction cup with a small audio plug that works tolerably well on most handsets and tape recorders. This set up is very inexpensive. One problem is that the suction cup falls off the receiver in mid-call, and the wire from it is cumbersome.
I never used the plug-in device you mentioned, but that should be all you need.
Note that in some states you must have the beep tone although that seems to be rare these days.