Could muscle weakness and rapid heart rate when standing up and out and about be due to adrenal fatigue? I've been feeling this way for a month.

Unlikely . Adrenal deficiencies are very uncommon. More benign causes are more likely. You could have low blood pressure and positional rise in your heart rate could be connected to that - the heart is trying to make up for a positional drop in blood pressure. This is called orthostasis.Also make sure you are well hydrated. Viral syndromes - usually benign and self limited - can cause a sense of weakness.