Either annually or yearly can and frequently does replace ‘every year’ as none of the phrases is limited by the number of occurrences, except to the extent that what happens twice a year is strictly biannual, not twice annually.
There is no difference at all among ‘annually’, ‘yearly’ or 'every year' and Longmans and Oxford Online don’t think there is. All their examples are almost wholly unhelpful but that doesn’t change any of their definitions.
Longman says annually means ‘once every year’
Longman’s example ‘The jazz festival is held annually in July’ is purely about grammar, not meaning. Test this by replacing it with ‘The jazz festival is held every 17 years in July’
Oxford says annually means ‘once a year; every year’ Oxford’s examples ‘the prize is awarded annually’ and ‘sales are increasing by about 17% annually’ purely about grammar, not meaning. Test this by replacing ‘annually’ with ‘every 17 days/weeks/months/years/centuries…’
Longman says yearly means ‘happening or appearing every year or once a year’
Oxford says yearly means ‘Happening or produced once a year or every year’
Your Dengue outbreaks seem very confusing. In my country your ‘rainy season’ does happen once a year; that's why it's called 'the rainy season'. We could be wrong and then so would you be, because more than one would necessarily be ‘seasons.’ If Dengue happens more than once a year then it happens several times or twice or however many times a year and certainly, that could and often would be expressed as ‘three times annually’ or ‘five times yearly’ or 'several times every year’ or '… each year' or '… in a year' or '… during the year'.
Usually more confusing are biannual, biennial and perennial none of which means the same either as each other or as annually/yearly/every year and about which again, Longman and Oxford agree. Their relevance here is to distinguish all of them from the examples in your question.
Longman’s biannual: happening twice each year
Oxford’s biannual: Occurring twice a year
Longman’s biennial: a biennial event happens once every two years; a biennial plant stays alive for two years.
Oxford’s biennial: Taking place every other year.
Longman’s perennial: continuing or existing for a long time, or happening again and again; a plant that is perennial lives for more than two years. Oxford perennial: Lasting or existing for a long or apparently infinite time; enduring or continually recurring