If your main goal is to lose weight you will be better off counting your calories and walking more.
Find out how many calories your body uses from just being alive using a calorie calculator and setting the activity level to the lowest possible then eat that number of calories minus 300 for a while and you should lose weight. The most effective way to track this is by weighing everything you eat (including cooking oil). It can be annoying at times, but it will be the most effective way to lose weight. If you don't notice your weight going down after a 3 weeks reduce calories again and keep repeating this process until you notice your weight going down. Another tip is to eat much plainer or bland foods, they will kerb your appetite however, I don't recommend this unless you are hungry all the time.
You said already that you walk your dog a lot which is actually really good. Simply walking more is one of the best ways to reduce calories as its really easy and not intense at all so you can do it all you like without getting that tired. Also if you have a dog it's probably quite fun as well so keep doing that if you wanna lose weight.
Running is also great and if you enjoy it go ahead however, it does tend to make you hungrier so watch out for that. But again, as long as you control yourself it shouldn't be a problem.
Weight training is very popular as a lot of people simply find it fun. You also pack on loads of muscle when you are a beginner which can really hook people in once they get going. It is quite good for weight loss in the long term as muscle uses a lot of calories idly allowing you to eat more and not gain as much weight. Keep in mind if you do go down this route you will need to track your protein intake (and calories if you want to lose weight).
A lot of fitness channels tend to discuss this quite well and give great advice. I would personally reccomend Jeff Nippard, Sean Nalewanyj and Renaissance Periodization (And for the love of god not Greg Doucette). Good luck!