I'm going to propose something slightly different. I think we all recognize that there is a series of words, glare, glaring, glaringly. However the parallel series of words, blare, blaring, blaringly results in a word that is not generally accepted as a word and is the word in question - blaringly.
The two previous answers, which both point out the visual nature of glare and audible nature of blare, are useful for that particular reason. However I don't think "it's glaring at you" is the right definiens to choose for this particular application, but glaring in the sense that it is reflecting (focused) bright light. With this interpretation glaringly would be the adverb of choice for visual events while blaringly would be the adverb of choice for audible events. I feel that both these words, glaring and blaring, have an inability-to-ignore connotation, but that is tangential.
For comparison, I put forth the words oblige and obligation; the former is a verb and the latter is a noun, both of which English received from Old French or Latin directly (approx. 13th century). In the course of verbifying the noun we received the word obligate (and it's relatives, such as obligated) which are now common words. My understanding is that the -ation suffix comes in its entirety through the Latin and French streams, and is applied, in its entirety, to the word oblige, resulting in obligation. To reverse this properly, one would return to oblige, but it appears that is has instead been shortened only partially to obligate. Thus, English now has the words obligate from the 16th or 17th century when in occurs in no other language, probably from this "back-conjugation" procedure.
That's all to say, what I'm suggesting is just a conjugation of blare or blaring to blaringly that conforms to other similar English conjugations, and would be construed as the natural development of language over time.
Part of the original question was whether to use "glaringly obvious" or "glaring." In the exact examples offered, I don't think there is a difference, but still have the following opinion: If the same choice was offered 7 times in a paragraph, I would, as a rule, choose the latter as choosing the former for all 7 would create an excessively wordy paragraph that didn't increase content in any way. That said, however, glaring is an adjective while glaringly is an adverb; this implies that at some point where the verb is not redundant to the adverb that glaringly will be preferred. For example, something could be glaringly obnoxious which implies that it is both vile and obviously so, which is not as redundant as being obviously obvious.