Modify your Java security settings to trust the certificates presented by your ASAs and/or ignore Cisco's code-signing issues. More details are in the ASDM Release Notes.
Default Java installation settings will indeed break ASDM. However, with tweaking, I use Java 8 on dozens of ASAs.
I keep my Java updates current via manual upgrades and am careful to reject Oracle's suggestion / default settings to "enhance" my Java security settings.