If you're always tuned to drop D, one of the issues you may be facing is that your E string (which is a D string) is too skinny. Most E strings, tuned to standard pitch, are already the loosest string. D-tuning just makes that worse.
Part of your issue may also be the cabinet - playing high on the e/D string gives you more fundamental than playing the same notes on the A string - it's how things work with typical pickup positions. If the cabinet can't handle the extra fundamental, you'd hear it more on the E/D string, and depending on various factors, having it show up on higher frets makes sense.