As I mentioned elsewhere, my life took a serious downturn, and I had to give up my cats, as I didn't know if I'd have anywhere to live myself. The first thing I did was contact the local rescue, and all but two of them were rehomed. It wasn't an easy phone call to make, but simply abandoning them would have been much more painful.
When I got settled in my new home, I was able to get two of the cats back. If I hadn't called for help, and just dumped them, I don't think I'd be able to have cats now. No reputable rescue would touch me with a bargepole, and I might have been prosecuted for neglect, quite rightly.
Nobody knows what's around the corner, but if you care about your pets their welfare should come, if not first (I don't have children, so they do with me!), at least pretty damn high on the list, and much higher than embarrassment or inconvenience.