Actually, a will is a specialized form of promise, a recorded promise, if you will, where upon the occurance of an event, property will convey from one person to another. While the details for any given reference should not be ignored, making too sharp a distinction between covenant and testament will tend to obscure more than it illuminates.
That isn't the case for a Covenant.
Heb.8 is speaking to Israel (note the words Israel and Judah in Heb.8:8) and isn't referring to our present salvation.
What is 'old' and done away is the old Mosaic covenant.
The Bible is the Old and New Testament, not Old and New Covenant.
Separating the two is a novel concept to me and not according to the real world or scripture (Heb 7-10 among others) and leads to heresy which always happens when basic Biblical truths are denied.