The pronunciation of this Old English word is difficult to express in text. The first element éa- is a diphthong equivalent to the vowel sound in English words like 'bear' or 'air'. (The use of 'ai(r)' in the pronunciation guide is an attempt to suggest this - vowel sound of 'air' should be pronounced, but not the final 'r').
The closing -ig is a common Old English ending pronounced as a long 'i' (that is, 'ee'). The final 'g' is not pronounced as consonant. This is actually the historical source of the modern English -y ending for forming adjectives (so for instance the noun 'luck' gives rise to the adjective 'lucky'). This example is in fact directly relevant to the word éadig, since éad was a noun meaning 'good fortune, happiness, wealth', so adding -ig made the adjective éadig 'fortunate, happy, wealthy'.