Wanting to change from bar end to road shifters
According to answers to this question, Shimano road and MTB had the same cable pull ratios up to 9 speed, but MTB 10 speed cable pull ratios were changed, which means that Shimano 10 speed road and MTB shifter and derailleurs are not compatible. I think that means that the pull ratio was changed for both front and rear derailleurs.

As you suspected you will need to replace front and rear shifters and derailleurs. If you want to keep the triple ring cranks you need a groupset that works with the  48/36/26 triple chainring and the 11-34 cassette. After a bit of searching around I'm sorry to say I think that will not be possible without changing gear ratios.

The only 3x10 Shimano road groupset is the 3x10 Tiagra, but the rear derailleur will accommodate only 20 chainring tooth difference, you need 22; largest rear cassette sprocket size is 32, you need 34. Total capacity is 41, you need at least 45.

If you swapped the smallest chainring to a 30 tooth it might work. I have heard that Shimano derailleurs will work with a large cassette sprocket 2 teeth larger than the specified max. Swapping the cassette and rings to Tiagra native 11-32 50/39/30 will of course work just fine.

Answers to this question indicate that Sram uses the same 1:1 pull ratio for a range of road and MTB groupsets. This page lists the 1:1 pull groupsets and it seems that Apex shifters are compatible with GX, X9, X7 etc. derailleurs. The X9 and X7 groupsets have 3x10 options but unfortunately SRAM offers no 3x road shifter.

If you can replace the crank and chainrings to 2x then you will of course be able to combine Sram Apex shifters with GX or X9 2x10 derailleurs. You would probably want to replace the cassette with an 11-38 or lower to preserve the low gearing. 

You are looking at replacing a significant chunk of the bike which you will have to figure out if you want to spend money on. Another possible cheaper option that occurs to me would be to go to a flat bar and fit Deore MTB shifters.