I get your frustration. I really do, as 16th century Portugal is exactly what I focus on. And yes, the amount of English language books (let alone other materials) on Portugal of the sixteenth century is limited. And the ones that do exist are mostly in topics relating to exploration and empire building, less in everyday life topics.

That being said, there is still a lot of resources. First a copy paste of the books list from my user profile of books I can recommend, but are more general in nature:

• Portuguese Seaborne Empire 1415-1825 (1969) by C.R. Boxer. Boxer is till this day the accepted authority and this book is still the reference book for the Portuguese empire from it's beginnings of expansion to the breakaway of Brazil.
• Prince Henry the Navigator: A life (2000) by Peter E. Russel. The ultimate biography of Infante Henrique in English, who was one of the key people in kickstarting the Age of Discoveries which lead to his elevation to mythical status during the 19th century. This book tries to reveal the 'real' Henrique and goes less into the details of 'his' discoveries (conducted by his authorization by other people) and more into his own life, motivations and decisions.
• Foundations of the Portuguese Empire 1415-1580 (1977) by Diffie and Winius. Detailed book detailing the beginning and peak of the Portuguese Empire in the 15th and 16th century. Accent on the events, battles and general military and political developments. Good also for general review of ships, navigation methods, weapons and equipment.
• Portuguese in Asia: 1500-1700 (2012) by Subrahmanyam S. Accent is on the economy and politics of the Portuguese presence in Asia. It is full of data and statistics about trade, shipping, earnings, costs and the general way the Portuguese presence in Asia was set up. Also gives slightly more information about the Asian surroundings (but the focus is still on the Portuguese).
• The Portuguese in India by M. N. Pearson. Focuses in entirery on Portugal on Indian subcontinent and their organization and administration.
• A History of Portugal and the Portuguese Empire: From Beginnings to 1807 Volume I and Volume II (2009) by Disney A.R. A great overview as well as introductory reading to Portuguese history. You can easily jump to chapters one is interested about. Volume I is chronological history of the mostly mainland Portugal, while Volume II goes into greater details about their colonial holdings.

I think of the bunch, the book Foundations of the Portuguese Empire 1415-1580 is most focused on the sixteenth century, and I can recommend it. Additionally Boxer's Portuguese Seaborne Empire 1415-1825 while covering a broader time period is really good at explaining in broad strokes the society and some institutions and organizations that one would deal with on a day-to-day basis in Portugal at the period.

I didn't list them, but there are also several books out there which cover some more minor details or minute events of the time, which might be really illuminating and more what you want to know. Some examples are:

• The Global City: On the Streets of Renaissance Lisbon by Annemarie Jordan Gschwend

• The Story of Suleyman: Celebrity Elephants and other Excotica in Renaissance Portugal again by Annemarie Jordan Gschwend

• The Calabrian Charlatan, 1598 - 1603 Messianic Nationalism in Early Modern Europe by H. Eric R. Olsen

You might also like to read some translated primary sources of accounts of the period, however they mostly relate to Portuguese exploration and I list important ones also on my user page. They are mostly available for free on archive.org and gutenberg.org as their copyright lapsed.

Other then that, I would hazard to say while reading about contemporary English might offer some insight into Portuguese life, it would be rather more prudent to try find resources on Spanish of the period, as they were culturally much closer to Portuguese then English, and will be a better reference point. For example I found the description of sailor and navy life in Spain's Men of the Sea: Daily Life on the Indies Fleets in the Sixteenth Century by Pérez-Mallaína much closer to Portuguese experiences then I found in English narrations of their 16th century sailors.

 

I myself, and as it seems most english speaking historians of portugal, am mostly interested in exactly the Exploration aspect of Portugal, as well as the Portuguese shipping and commerce with Asia. For that my resources are adequate, and with those I am most familiar with, but if you can help me narrow down what you are interested in I might be able to recommend some more readings and articles.