Going from Tourney to anything else will not solve your shifting problems if your cables, housings, shifters, etc. need work due to worn or poorly-installed parts (likely cables) or poor adjustment. Unless you find and cure the root cause of your troubles all you will do is to lighten your wallet.

First examine the cables inside your brifters to make sure that one has not begun to fray; getting a broken-off cable head out of a Shimano brifter is exquisite torture. Always check this when any sudden change in shifting is noticed.

Then remove the shift wire from the derailleur clamp so that you will have slack to be able to thoroughly examine your shift cables, both inner wires and housings.

Check with your derailleur documentation to determine the correct routing of the wire to the clamp bolt for when you put it back; the correct routing is usually OVER the little tab next to the screw. If it is routed wrong it will throw off your indexing. This is a frequent mistake!

I would then start by scrutinizing at the last loop of cable going into the rear derailleur, it is a frequent cause of shifting woes, moving along to the others systematically. If there is any doubt as to their condition I would suggest that you replace them all, with lined housings and die-drawn stainless wires, they pretty are cheap. Then move along to check your levers and derailleurs for free and correct movement. Then reinstall any removed/disturbed components and reattach the shift wire CORRECTLY.

When all is well with the hardware I would suggest that you follow this procedure from the beginning without skipping any steps, as if it were a totally new installation: Park Tool Co. » ParkTool Blog » Rear Derailler Adjustments (derailleur)