It's completely understandable to be a little apprehensive about having to engage higher-level analysis of sources under time pressure, but don't worry! There are several steps you can take to help you get in the habit of source analysis. It's a skill you can develop like any other. 

First, read the source all the way through. Before trying to apply it to the question or pick it apart, take a moment to try to literally explain to yourself, or note down, the 'message' of the source, in a single sentence. Ask yourself how the author of the source would summarise his or her 'argument' in simple language. This can be especially useful if the syntax is not in modern English, and can form the foundation of your analysis whilst calming your nerves.

Next, use coloured pens to underline certain words which correspond to parts of the question. These might be lines of argument or examples. 

When dissecting other sources, use the same colours to easily 'cross-reference' between sources. 

From this you can build your analysis. Look for differences, but more subtle may be the similarities between the sources. 

In the sources you should find references to your background own knowledge. Try to resist the natural urge to overpower your own argument with your own knowledge; it is likely the question is more curious about your ability to analyse the sources on their own terms.

Keep practicising, and try to maintain a natural curiosity about the process of investigating the sources; this will come accross to the person marking your work.