Hi andfam33 and welcome to the forum! Thanks for taking the time to post your problem, I’ll do my best to help you out. Oh, and sorry for being a bit slow getting back to you.
First of all, if that’s all they told you then you should ask for more information and dig deeper as they aren’t being specific enough. It’s too simple for them to say that its “going out” and that it would be too expensive to perform any minor repairs. Is it a problem with the bearings or the seals? Did they check the fluid level? A more detailed diagnosis is needed.
Next, it’s important to know that the transmission and the front differential/drive axle are all in one unit. So in order to fix the problem, the “transmission” will have to be repaired/rebuilt to a certain extent. If it is only a minor problem with the bearings or seals then a replacement isn’t necessary. If the damage is any more than If the symptoms you’ve noticed (whirring, whining, rumbling, etc.) have been going on for a long time and/or have become more severe, the gears could be quite damaged at this point. If in fact it is anything other than a minor problem, a replacement is probably the safest and most cost effective option.
As always, my advice is to get at LEAST a second opinion, if not a third from shops you know you can trust. Get estimates from each of them, both for the cost to repair the differential and to replace the entire transmission.