2004 Honda Accord LX, 4 cyl, AT Electrical Issues

Hello all,
I have a 2004 Honda Accord with a recently installed used engine. The car ran fine for about 1000 miles and then started experiencing some major electrical issues. Battery is about 1 year old and alternator was tested good last fall at Auto Zone. Car has sat since December as there has been no need for it. I’m now trying to get it running as it’s become a driveway eyesore. The car drives fine until the gauges start going wacky and then the car starts to run crappy until it dies. Todays experience……….charged battery fully to 12.68 volts. took for a test drive for about 5 minutes. Batt. was 11.10v (car running) after test drive. Batt. read 11.63 after shutting engine off. Upon start up, batt. light stayed on for 6 seconds, and there was a relay clicking sound. The relay clicking sound and even flashing marker lights have been occurring since this problem started. The car does have a remote starter installed, which may have something to do with the relay clicking/flashing marker lights (just a guess). Anyway, if I didn’t have the alternator tested last fall, I’d say it needs replacing. All connections are clean and secure. Could this be the computer failing?