.. is there a way to do this besides having to give them a "discount" everytime? I'm afraid they'll see my new price and look elsewhere.
If you're concerned your repeat clients would look elsewhere, then maybe the solution is to go to past stays and send a friendly message to each one that although you're raising your rates, you're glad to keep their bookings at the same rate and you will make a special adjustment on each booking from them. Then, post the increased rates. It seems the only way to accomplish what you want will be to make a Cost Adjustment to those repeat bookings.
I've raised my rate by $5 since starting on Rover and thus far it has had no effect on my business with repeat clients. I even had one client take note of it and praise me for doing so, as my services were "well worth it." If you've proven yourself to be dependable and meet or exceed client's expectations, most of them will not be concerned over an increase of a few dollars. After all, it is a small price to pay in order to know they're leaving their companion(s) with someone they know they can trust. Generally speaking, those who are concerned about every additional cent you are charging them are likely to be the most high maintenance clients of all and you may or may not want their business anyway.
I like so many Rover sitters I started with a low rate to attract clients and that worked very well. I raised my rate for new clients mid-year pretty significantly and quickly booked two new clients at the new rate; by the way I am a daily walker and have nothing but repeats. I told my existing clients I had raised my rates via text and that they were grand fathered at their original rate, so I adjust the booking.
I just informed my entire client base I would be raising rates effective 1/1/17 by $2 per walk, so far zero fall out or push back. If I lose one or two, that opens me up to book new clients at the much higher rate.
Never cut your self short, set your rates competitively, and not just what you find on Rover, but check the local independent companies to see what they are charging.