"Водолазовой" here is an adjective that comes in pair with word "службе":
Ты ведь не на службе / Живешь водолазовой. = Ты ведь живешь не на водолазовой службе.
Водолазовая служба literary means diver's service.
Why Esenin is using this metaphor is not clear for many people - there are a lot of questions about it on many internet sites. In the book "Неизвестный Есенин" the author proposes a version about a story that could be known to Esenin about a diver that gone mad when found a "forest" of drowned people (killed by bolsheviks). Another possibility is that Esenin says "don't dive into the depths of my soul". Or maybe Esenin thinks that he is "drowned" and asks the diver that wants to rescue him to go away.
As for spell checker suggestions, there was a known bug in MS Word's spell checker a couple years ago that made funny suggestions for "wrong" words. For instance, it suggested to replace the word завязочками (like in "папка с завязочками") with "завяз очками" (got stuck with eye-glasses). All words with "-очками" ending were suspicious and many gave funny replacement suggestions.
So, spell checker is not always right, especially when the words are rare or the text is rather old or poetic.
Looks like version that водолаз is synonym for священник has many supporters. For example, here the priest's youtube channel that is called Служба Водолазова.
On Gramota.ru forum the following citation is posted:
У Макаренко, в "Педагогической поэме": "А вот пусть Осип Иванович скажет: для какого хрена водолазы здесь это, понадобились?", когда пришли сватать колонистку и говорили о церковном венчании.
...Про водолаза говорит не воспитанник колонии, а работавший там Силантий, самый что ни на есть сельский трудяга, и говорит он простому крестьянину, причем оба друг друга прекрасно понимают.