Either of those shifters will work. But the Aceras are combined with brake levers so you'd need to redo the brake cables as well when installing them. I also prefer trigger shifters like these over GripShift ones.
The chain slipping sensation could be a result of a stretched chain, worn cogs, faulty adjustment of derailleur, faulty derailleur, misalignment of derailleur hanger, or a problem within the ratchet mechanism within your freehub.
Measure your chain with a ruler first to determine stretch. Next, take a clear closeup photo of your rear cog teeth from the side and post it here. People on BF cannot resist some good gear photos.
After that, you still may need to do some more troubleshooting and website research.