Does myfitnesspal double up on energy expenditure?

The daily activity increases in calories covers increased daily activity - not exercise unless your workouts are only walking for not long.
Then it might - by sheer coincidence.

It's better to get the activity level correct for what you are doing for non-exercise time.
You got a family and household duties and a desk job? Lightly-Active

Desk job & sitting commute and come home and plop down on the computer for rest of the evening till bed? Sedentary

Log the workouts accurately for time and intensity if there is an option (speed or pace).
If very low key workout like walking - take half the time or half the calories.
If no intensity description, perhaps 75% of the time or calories given.

So it really depends - if your daily life really was active outside exercise (and some are) - then not double counted.
If it sounds like the way you did describe it - probably some overlap.

But get the activity level right - don't pick a totally incorrect number for exercise like 0 calories.
You need decent stats to look back on to figure things out - purposely doing things wrong won't help that situation down the road.