Is the failure of the one making dhikr to say, "My Lord, indeed adversity has afflicted me, and You are the Most Merciful of the merciful ones" – because what afflicts him is not a major issue compared to the calamities of others – considered correct, or is it excessive sensitivity and shyness from God?

What you are doing by avoiding this supplication is not correct. You should humble yourself before Allah the Almighty and supplicate for the removal of your distress, the alleviation of your anxiety, and the lifting of harm from you. Allah loves for His servants to ask Him, supplicate to Him, and humble themselves before Him, as He commanded in His saying: "Call upon your Lord humbly and secretly." So, supplicate to Allah with this noble supplication with which Ayyub (peace be upon him) supplicated, and with others. For supplication is one of the greatest acts of worship and the most excellent means of uncovering afflictions and repelling evil.