No, not every Korean word needs 받침 (patchim or batchim). Many words don't have it, like 나라, 하루, 아저씨, etc. When spelling foreign names in Korean, 받침 is used when necessary but their use is not encouraged. The reason is because certain 받침 don't retain their original sounds unless they are followed by a vowel. When is 받침 necessary? It is when you have two or more distinct consonants in a row or have one at the end of the word. For example, in Hancock(핸콕), the n-c sequence turned "n" into a 받침, and the [k] sound at the end became a 받침. Your name Ajané has no multiple consonants or one at the end, so there's no reason to use 받침. It's just 아자네. Also, even consonant groups and end consonants don't always become 받침. It is a complex subject, but certain 받침 situation can be avoided by adding the neutral vowel "_" as necessary. For example "Tod" is usually spelled 토드 adding a syllable using "_" instead of 톧 with a 받침. This is because 토드 pronounces closer to the original sound than 톧.

Well, Let me put it this way. Both 아자네 and 앚아네 can be used because they pronounce the same. but the question is when to use what, right? more common way to do it is the former rather than latter. I don't have a handy explanation as to why that is the case, but in my opinion, I think I am doing the former one so that I don't need to have 받침(Pat'chim) because that feels like more of a basic. Why bother making it difficult using 받침 if you can use the easier one without it, right? Hope this helps,