What were the anxieties Spenser had over the course of his poetic career?

Edmund Spenser lived during a period when England had finally emerged from a long period of internal wars, religious transformation, and uncertainties about the legitimate monarchy that would emerge. Elizabeth I not only stabilized the situation, she made England into a world power by carrying out a successful war against Spain.
One can understand Spenser mainly by considering this historical context. No one in England at that time could be sure that the country would retain its stability and not possibly be defeated by a foreign power (Spain) or lapse into religious chaos. Spenser was a devout Church of England adherent. In The Faerie Queen, he presents an allegory of what, for his time, was a desperate struggle of the true (in the Anglican view) church against its enemies. Redcrosse is prophesied to be the champion of this new, stable religious realm: "Thou Saint George shalt called bee / Saint George of merrie England, the sign of victoree." His defeat of the Dragon symbolizes the triumph over the false church.
Other works by Spenser reflect a similar concern about religion, typically expressed in allegorical terms. Astrophel is an elegy on the death of the poet Philip Sidney, who was killed fighting the Spanish in the Netherlands in defense of the Protestant Dutch. Spenser also obsessed about the need for Ireland to be fully "subdued" by the English. The Irish adherence to Roman Catholicism was seen as a huge threat, and, unfortunately, Spenser expressed his fear of it in a manner which set the tone for, or at least encouraged, the centuries of oppression and violence the English imposed upon Ireland.
When not focused upon religion, Spenser poetically expressed his anxieties about the transient form that human emotion can take, namely, the vulnerability of love and devotion. "One day I wrote her name upon the strand / But came the waves and washed it away." This sonnet is probably his most famous one from Amoretti. It expresses both the fear of the ephemeral nature of earthly life and also the reassurance, by his mistress, that through art and through poetry, his love and fame will be assured.
In his personal life, Spenser was fortunate to have been granted a stipend by the queen. As a result, the normal anxiety everyone in the arts has about where the money is going to come from was at least partly relieved by the woman whose greatness he celebrated so vividly in The Faerie Queen.