There were several versions of the LXX and the NT writers cited whichever version suited at that moment. And they were not beyond stretching the meaning of the text to suit the point they were making.
I do not know if Jesus used the Hebrew text or the Greek, although one would need to be conscious that he was an Aramaic speaker and he preached in a very small area which had experienced limited Greek influence. Jesus did not write anything nor did any of his immediate followers. The earliest written record, the 7 genuine letters of Paul, do not recount Jesus' life or preaching. The earliest Gospel (Mark) was written 40 years after Jesus' death (and borrowed material from Paul). Matthew was written some 15 years after Mark, while Luke was initially written about the same time buit the version we have dates from the early second century (along with Acts, which is also a second century document).
The MT is about 1000 years younger than the LXX. As Emanuel Tov (obviously a Jewish scholar) points out ("Textual Criticism of the Hebrew Bible"), the Jews have one Hebrew text because the Masoretes (about 1000 years ago) created their text and destroyed all variants. He says that in the process, the Masoretes set in concrete all of the errors.
It is estimated that more alterations have been intoduced to the NT text (deliberate and accidental) than there are words in the NT.
The use of ancient Hebrew characters for HWHY into the scrolls, characters that had long ago ceased being used is something that the WTS does not do. Instead, it adds - wherever it suits them - the Roman Catholic word which amalgamates YHWH with Adonai (the "J" is soft in Jehovah, as in "Yahwist"). The paleo-Hebrew characters were inserted a long long time after the scrolls were written - perhaps looking back into their history.
Prior to the 6th century neo-Babylonian era, the Hebrews were polytheistic, including the worship of YHWH's wife, whose name was Asherah. The monotheists did not win out until the return from Babylonian Captivity, and all that we read is their propaganda. 95% of the population was illiterate -- and remained so until very recent times. At least one of my grandparents was illiterate.