He does, in fact, use more than one Name. And the variants are VERY important, in context, and often express a title in a specific situation. (Yahuah Nissi - our "Banner", YHVH Rapha - our "Healer", etc. You can easily find books on those.)
"Bereshiet bara Elohim..." in Genesis 1:1 is the first such, and later in Genesis His Name (aka 'the Tetragrammaton', YHVH in English letters, or Yod-Hay-Vav-Hay) appears -- first in Bereshiet/Genesis 2:4. He is also (primarily) known to the patriarchs as "El Shaddai" (which can be rendered several ways, but I prefer "the All-Sufficient El") and then in Exodus Moshe is told the patriarchs knew Him primarily by other names, but as YHVH, "I did not MAKE MYSELF known to them." Even so, it was a Name they DID know, as in Genesis 28:16 when Yakov see his famous ladder (which makes v 20 interesting.)
'Elohim' (which is in fact plural, albeit the 'Royal Plural' in His case arguably) is more a title than a Name; often rendered 'God' - although the same word, usually plural, references pagan "mighty ones," too.
I suggest the best key to understanding the context and importance of His Name, as Written, in that proper context, is do a word study with a good computer tool or concordance (BLB, E-Sword, etc) for "ki ani YHVH."
It is, in fact, THE overriding theme in the entire Book of Exodus. 'You shall know, THEY shall know, Pharaoh shall know, Mitzraim (Egypt) shall know, ALL shall know'..."ki ani YHVH". That My Name is Yahuah.
Later, in SO many of the prophets, we are told something I contend is VITALLY important, and just plain not taught in what I thus refer to as the 'whore church' which has changed His Word, virtually done away with His Name, and ignored the prophetic significance of all that is now in progress:
"Then they shall KNOW that My Name is Yahuah."
It does NOT say - EVER - "My Name is 'the LORD'" But it does say we will know His Real Name.
Do a search for yourself, and see how often that theme and prophecy is repeated. AND the context.
(Jeremiah 16:21 is one place to start, there are many.)