I recently got back into martial arts so I can relate. A few things I've learned:
1) If doing MA and weights on the same day, try to separate the sessions by a few hours. This may depend too on the intensity of your MA training. I train in private one-on-one sessions so there's no downtime. I walk out wringing wet with sweat and barely able to move. If possible, don't do both on the same day. I currently have a mix: sometimes I train at noon with weights and at 6 go to MA. Ideally, I don't do both on the same day but it happens.
2) Adopt some Chad Waterbury style weight training - don't train to failure, don't waste too much time on direct arm and shoulder training, stick to big movements, short workouts, explosive concentrics, don't seek fatigue and soreness, etc etc. You may be surprised that when you put weight training "on the back burner", you actually begin to grow.
3) Consider a post-workout drink after weight training AND MA training. Take advantage of other recovery tricks as well.
4) I cut out most gym GPP/cardio work after starting back in MA. The MA takes care of it. I do one session per week of pure GPP now instead of 2-3.
5) Adjust diet as needed. Again, depends on the MA training. Are you just farting around with a friend or are you really getting your ass kicked for an hour or more?
6) Don't give up weights, but focus on MA for a while if you're serious about it. If a man chases two rabbits, both will escape.