There is a problem with answering this question, and it lies in the fact that Jewish sources prior to Jesus but after the translation of the Pentateuch into Greek are fairly sparse. There are some Greek texts from Qumran, but the number of Greek biblical texts is negligible. Emanuel Tov has an article on this, called "The Nature of the Greek Texts from the Judean Desert". In case you don't have JSTOR membership, I have downloaded it for you here. As you can see, all that the existence of Greek texts from Qumran can tell us is that people spoke Greek. Prior to the adoption of the Septuagint by the church, their using a Greek translation was hardly controversial.
To the best of my understanding, it is generally reckoned that Greek biblical texts were very common amongst Jews throughout the Greek-speaking lands of their dispersion (particularly in Egypt), and there is no reason not to suppose that this popularity prevailed also in the land of Israel. Mel Gibson's movie notwithstanding, the lingua franca of the region at that time was Greek, not Aramaic. There were Aramaic translations of the Pentateuch as well, but they are understood to have served as supplements, rather than replacements, and (like Hebrew texts) were targeted at scholars, rather than the laiety.
So far as the latter part of your question is concerned, you are entirely correct, but the displacement of the Septuagint by Jews happened over the course of several centuries. The earliest rabbinic sources without exception all present the Septuagint in glowing terms. In the Mishna, Megillah 1:8, Rabban Shimon ben Gamliel is quoted as having said that Greek is the only language, other than Hebrew, in which it is permissible to write Torah scrolls (sifrei torah). Assuming this to be historically accurate, the Gamliel to whom it is referring was either a contemporary of Jesus (and the person from whom Paul claims to have studied), or the grandfather of the same. Either way, his son Shimon (and the person to whom this quote is attributed) is a rough contemporary of Jesus himself.
Commenting on this statement of his, the Jerusalem Talmud (Megillah 71c) says that the sages checked and discovered that Greek is the only language into which it is possible to translate the Torah with its exact meaning, while the Babylonian Talmud (Megillah 9a) presents an origin story for the Septuagint that testifies to the miraculous nature of its composition. This origin story is very similar to the earliest extant text that testifies to the Septuagint's composition: a pseudepigraphic Jewish letter that purports to be written by Aristeas, King Ptolemy's bodyguard.
As you may be aware, the story involves no fewer that seventy elders, sequestered in seventy different rooms, with no foreknowledge of the reason for their being brought, animated by a spirit of prophecy and translating the Pentateuch identically to one another. (If you look up the passage in question, it presents a fascinating array of the changes they are said to have made to the text, and the reasons for those alterations).
By the time we come to the composition of the minor tractate, Sofrim (perhaps composed as late as the 8th century CE), Jewish attitudes towards this text had changed rather dramatically. There, in Sofrim 1:6-7, it is explicitly said that sifrei torah cannot be written in Greek, that it is impossible to translate the Torah into any other language without changing its meaning, and that the day on which the Septuagint was written (by no more than five elders!) was as hard for the Jewish people as the day on which the golden calf had been constructed.
The reason for this change of attitude lies in the fact that the Septuagint (which, in time, came to denote the translation into Greek of the entire Bible) was embraced by the nascent Church and used in anti-Jewish Christian polemic. A favourite in that regard was the translation of Isaiah 7:14, in which עלמה ('almah, "young lady") was rendered παρθενος (parthenos, "virgin").
By the time of the composition of minor tractate Sofrim, Christians had composed a number of anti-Jewish tracts that utilised the Septuagint for the purposes of demonstrating Jewish stubborness in their rejecting Jesus. Perhaps the most graphic of these texts was a 6th century text entitled Disputatio cum Herbane Judaeo: "Disputation with the Jew, Hurban". [The name is from the Hebrew word for destruction: likely a reference to the temple's eradication]. The author identifies himself as Bishop Gregentius of Tafra, in Yemen, but is only adopting that persona for the purposes of the text.
In it, he attributes the following sentiment to his invented Jew, Hurban:
Our fathers wrongly and capriciously translated the books of Israel into Greek so that you could take possession of the same and silence us.
With such attitudes as these, it was no wonder that the Jews came to view their Greek translation as a tragedy, to rule it out altogether as a liturgical text, and to undermine the earlier texts that spoke of the miraculous nature of its composition.
If you would like information, Martin Hengel's book is very good:
Martin Hengel, The Septuagint as Christian Scripture: its Prehistory and the Problem of its Canon (ed. David J. Reimer; Edinburgh & New York: T&T Clark, 2002)
