Congrats on your new betta! :D
Yes, I would definitely recommend getting a new heater. Usually it's best to keep the temperature at 76-80 degrees. I keep mine at 78. 60 is to cold. :(
I personally recommend the Eheim Jagger 25w heater. I've used it first for my 3gallon, and now for my 5.5 gallon, and it works like a charm. There's also the Aqueon submersible heaters which supposedly have a good rep? They have life time warranties. I think they may have a 25w?
Those two were recommended on this thread about heaters:
The thread creator has a 10g, so you'll of course have to go for a slightly smaller heater than what they were looking for. (Instead of a 50w, a 25watt is best for 5gallon).
Good luck finding a nice heater! :)