What is the sound being added between the words? It sounds like a short "oon".
It's the nominative indefinite ending, -un, which is rarely written, but if it were, it would be written as زوجٌ
Notice how the small symbol above the ج looks like two small وو linked together. The grammar behind when we use what when talking about indefinite things is a bit too involved to explain here, so what I suggest you do is pay really close attention to what you hear while following the Arabic letters with your eyes, and you'll eventually get a feel of what to use when. It takes a long time to acquire this skill, so just be patient and listen to a lot of Arabic while following text every day. You'll get there eventually :-)
PS: We also often drop the sound when reading out loud, so don't worry too much about it.
When جَدِيد is used with people, doesn't it also mean young? If so, when would this mean young versus a new one? Also, when you're saying "a new husband", do you mean somebody recently got married, or they're a divorcé(e) that has remarried?
Nope, in Arabic, جديد only means "new" and not "young." It can mean both a new husband as in somebody that a divorcé(e) has gotten married to, or somebody who has just become a husband.