Perhaps you should try resistance exercise that builds lean muscle instead of bulky muscle. Something like pilates. Or if you are using hand weights go for more reps with a lower weight rather than standard reps with a heavier weight. Are you also doing cardio work to burn fat quickly? Increased muscle raises your resting metabolism, but you have to check you're not also increasing what you eat to compensate.
To answer the question of whether weights are essential, weights as a tool aren't necessarily essential but strength training in general is highly advisable. You don't need weights for that, you can use your own body weight like with squats, push-ups and tricep dips. The chief reason you need to build muscle is that when you lose weight you lose both fat and muscle. So it's not a matter of building more muscle in many instances, it's about keeping what you already have.