I'm 28, fit and healthy, but I still use a push-cart. Not because I can't carry, but because it helps my game and it means I can play more golf. I've been playing golf for over 10 years and bought my first cart a couple of years ago. I've got a lot of rounds of carrying behind me and never had any problems with it. I preferred carrying, since it meant I could walk freely all over the course with my bag.
My body is a lot better on the back nine and after a round if I'm pushing instead of carrying. This means that my swing isn't as affected by the walking compared to if I'd carried. I also like to fill the bag with balls, tees, clothing, water and stuff. If I'm pushing, the added weight doesn't matter much.
The main reason I bought it was so I could play more golf. Seasons are short here and I want to get in as much golf as possible. If I push, I'm more likely to play multiple days in a row. Once I got the cart, I also noticed how much difference it made to my fatigue during the round. I feel less worn out and more ready to play golf towards the end of back nine.
I would recommend that you play a round or even multiple rounds with a push-cart or pull-cart and see how it feels. It is an investement, especially if you get one of the higher end carts, and it would be better if you were able to experience the difference yourself before making the investement. Your local course may have rental carts, maybe you got friends that can lend you one for a few rounds.
I can only speak for myself, but I noticed the difference from the first round I played with a cart, so you probably won't have to play weeks of golf to see how it works for you. Although it could also have a more of a long-term effect on your shape.
TL;DR: Try it with rental or borrowed carts a few rounds.