I think you are on the right track here. Get a pro bike fit and work on your core. Leaning on the bar is usually a symptom of fatigue. If the bike doesn't fit 100% fatigue sets in earlier. And yes, the core is where the weakness is.
In addiition:
Also ask for checking your bars when you work on the bike fit. A stronger bend might be more ergonomical for you.
Ergonomic grips (e.g. ergon GPs, GCs or GRs) might help you, too. They simply prevent your wrists from turning down.
In addiition:
Also ask for checking your bars when you work on the bike fit. A stronger bend might be more ergonomical for you.
Ergonomic grips (e.g. ergon GPs, GCs or GRs) might help you, too. They simply prevent your wrists from turning down.