Sounds like the alternator has some sort of lag. It should react to the load changes almost instantly so it sounds like the voltage regulator (in the alternator) is faulty. Is it a brand new alternator or a rebuilt? Of course loose or corroded wires from battery to alternator will create problems too so be sure those are clean and torqued down well.
Had what sounds like the exact same problem. It went on for a long time. Had Lexus check it out and they said it was 'normal'. I new it wasm't. Had many mechanics check variuos things, but know one could find the problem. Battery was good ect. Then one day by accident, I was checking the battery agian and took off the battry clamp (I think it was the '+' side) and found that the clamp was accually cracked right though. You could hardly even see it. The clamp would tighten up to what you think was a good tight connection to the post, but it wasn't. A new $2.00 clamp at Wal-mart and 5 minutes, the problem was fixed. What a drag for such a liitle problem Hope this is the case for you.
As for the shifter problem. It could very well be the selinoid switch beside the shifter. I relpaced mine a few years back. You should here a faint 'click' if you put you ear near the shifter and press the brake with you hand. If you don't then the switch is likley the problem.
Hope this helps.