I know this subject about hobby/burner phones has been discussed at length however curious to know that given the newer sophisticated app's available for personal cell phones such as Burner that allows one create alternate phone numbers to text and make calls out on using their main cell phone wondering if these apps are a good substitute vs. having a separate hobby/burner phone? Are these app's as safe and private as having a separate phone?
No.
If the texts and calls are (and they are) made over your data line, there's still an IP address and that's still traceable. Not by your girlfriend or your boss, probably, but if the cops want it the cops will get it.
It isn't truly anonymous. Now, I haven't heard of even the cops in the States getting that serious about prostitution. They only get really serious when you start wiring money over state lines for this sort of thing. Pay with cash and carry a hobby phone that isn't traceable to you and you're in really good shape unless you get caught in the actual act. Stick to known, reputable providers and you should be fine.
Think of it this way: if you got a hobby phone and used your home's WiFi connection to book appointments, wouldn't that be traceable by the cops? What's the difference between using your regular phone and using a burner app through that phone?
Seriously, if you're too cheap to buy a ~$20 hobby phone and spend ~$50/month (and there are cheaper alternatives to each of these options) to keep it going for the sake of protecting your anonymity, I'm hoping you don't have much to lose.