As for dispensations (or periods of time mentioned in the Scriptures): Well, I believe the Bible clearly teaches that there will be a Tribulation, and a 1,000 year reign of Christ; And that both believing Jews and Gentles will be present during both of those time periods (that are yet future). After that, there will be a New Heavens and a New Earth were all believers will reign with Christ for all eternity where there will be no more pain, sorrow, death, etc. If you are looking for Scripture on these different periods or ages of time, I can provide that for you (If you are interested).
As for believers today making a separation between the Old Covenant and a New Covenant: Well, think of it is terms of the Old Covenant as an apple seed that gets fulfilled into becoming an apple tree (i.e. the New Covenant). For Jesus said he came not to destroy but to fulfill. For if I destroyed an apple seed I would take a hammer and smash it. But if I took that apple seed and planted it in good ground and watered it properly, it would then have the potential to fulfill it's intended purpose by it growing into an apple tree.
For there are many things we can learn from the Old Testament (Like the confirmation of the New Testament teachings within the Old, the origin of things, and the Typifications of Christ, and seeing how God's entire plan of salvation was to be played out from the Old to the New). However, that said, I do not believe the Old Testament Law of Moses is still binding for us today, though. Based on Scripture, I believe a Christian obeys the Law of Christ or those laws set out under the New Testament. How so?
Well, for one, there are 1,050 + New Testament Commands and only 613 Old Testament Commands (Which would make it very complex to obey both of them). Two, many laws have changed when Jesus came (So the New Testament Commands actually conflict with the Old Testament Commands in certain cases). For Jesus changed the Law of an eye for an eye and said to instead to turn the other cheek. Peter was told to eat unclean animals by God in a vision. This was a change in the OT Law. Also, when Jesus was crucified, the Temple veil was torn in the Temple; Meaning, that the animal sacrifices were no longer necessary because Jesus is now our Passover Lamb now. For Christians are not seeking to rebuild the Temple today so as to sacrifice animals again. The animal sacrifices were just shadows or types of Jesus Christ. This animal sacrifices were a part of the Law in the Old Testament, but are no longer necessary in the New. For no man puts old wine into new wine skins otherwise it will cause them to burst.
Third, Scripture actually states that the New will replace the Old.
Check the many articles on this point of topic here at Grace Community International (Starting with this article):
https://www.gci.org/law/lawmoses
For GCI helped me alot to understand the difference between the Old and the New.