An unpaired t-test uses the difference between the means of the control and test datasets to determine the p-value. A paired t-test combines the control and test data first, by taking the difference between control and test values for each individual experimental unit, and then compares the mean of those differences to the theoretically expected value (i.e. null hypothesis value). In other words, the paired test constructs a dataset of differences and then does a one sample test.
For your circumstance the experimental units are the subjects and the control and test conditions are drug and no drug.
The advantage of the paired test comes into play when there is variation that is shared across the control and test situation, and thus can be removed in the within subject differencing.
The reduction in variation by pairing gives more power to the paired test, but at the cost of sample size. Twenty observations from ten subjects measured in two conditions gives 9 degrees of freedom for the paired t-test, whereas 20 measurements analysed in an independent arrangement gives 18 degrees of freedom. That means that if there is no shared variation to be removed by the pairing, the unpaired test has a little more power. However, you do not need to reduce the variation m uch to overcome that small power advantage.
In most cases if there is a 'natural' pairing between the control and test measurements then using the paired test is the best approach. Your situation is nearly as straightforward as a before and after comparison within subject and so the paired test is going to be sensible.