Okay! I took a quick look at your solution and here are some tips:
(Disclaimer: My solution shown below might not be (close to) optimal either since I just took a quick look!)
1. You should get better at doing DRs. 18 moves is a lot for DR. For your EO I quickly found 2 different 14 move DRs:
DR1(on white-yellow axis):
Your EO
(D' L) //4c4e
(B2) //creates a second 1x2 "bar" on DLB corner
(D R2) //creates a 1x3 bar on FD position
(B2) //creates a second 1x3 bar on UR position AND sets up the to correct e-slice edges to be on the same side
(U2 D' L) // solves the rest.
DR2(on red-orange axis):
Here you already have a 4c4e case after doing your EO
(L' U2) //creates a 1x3 bar on UF position
(R2 B2) //creates a 1x2 bar on URB position
(L' U2) //creates a second 1x3 bar on UB position
(L' R2 D) //solves the rest
And additionally there was even a 16 move DR using your 9 move 4c2e solution:
DR3:
Your 4c2e
(D' L2 D U) //setup to trigger
(L D2 L') //trigger
Take a close look at those examples above and make sure you learn how to more efficiently setup the triggers. Also only try to generally use 2 moves max to get from EO to 4c2e or 4c4e cases. If you cant find any 2 move or less setups you propably should try another EO.
2. Speaking of EO. Make sure you find optimal EOs AND make sure you check multiple different EOs:
I dont know if you saw it and checked it but there are some 4 move EOs on inverse on different axis.
(R D2 B2 D) and (R U2 F2 D) are both 4 move EO solutions.
Also it is VERY important to check many EOs. Most of the EOs dont have a good continuation so you should check as many of them as you can.
3. HTRs:
You are correct that your DR solution is actually 3 qt away from solved. The reason you got fake HTRs is because you used 2 or 4 or any even number of quarter turns to get to your (fake) HTR. To get real HTR you need use odd number of QTs since you are an odd number away from solved. You should google "hyper-parity". There is a document from which you can learn to quickly recognize how far (in QTs) you are from HTR and how you can move closer to HTR. That is vital in finding HTRs. How ever that is quite advanced so I would recommend you getting more experienced and comfortable with finding efficient DRs first and foremost.
Here is however a HTR I found for your start quickly:
First we notice from the hyper-parity graph that we are 3qt away from solved (bars+bars case, we could "solve it in 2 moves but that would give us a fake HTR since we have parity). We also immediately notice that just doing a U will move us to solved+one face case which is only 2qt from solved so we do that.
(U') //puts us 2qt away from solved.
Now we look at the hyper parity graph again and see that next we need to get to slashes+oneface case if we want to finish in 2 quarter turns.
(L2 U') // puts us 1qt away from solved
Note that you also want to have exactly 2 M or S-slice edges on the correct slice since we want to solve the edges in 1 quarter turn next along with corners. R2 instead of L2 would work for corners here but not for edges. And that will always be the case here: turning the opposite layer will have similar impact to corners but different impact to edges, which is very useful to understand.
Now we just want to create two 1x3 bars AND have the correct edges be in the "same side" of the cube once we are done. Just like when solving 4c4e DR case. Except we can only use half-turns now. So lets imagine we are trying to just do DR on blue-green axis. We already have two 1x2 bars and just doing R2 would make two 1x3 bars. But the problem with this is that it would leave the 2 edges (white-blue and green-yellow) on different sides of the cube. So we must first do:
(B2 L2 B2) //moves the edges to different sides of the cube
Now that the (whiteblue and greenyellow) edges are on different sides of the cube we can do
(R2) //creates two 1x3 bars while moving the 2 edges to the same side of the cube.
Rest is easy. Just finish like it was DR.
(D2 F2 D) HTR
Now dont worry if the HTR stuff seemed too complicated. It is hard at first but getting better at DRs helps. And learning the hyper-parity graph is essential.
I hope this helps you to get better and understand some concepts
Edit: link to hyper-parity:
https://docs.google.com/document/d/1Onc2lu5iW_ODnT71oN5lgpVVjVqbvUMzaylzxgykx0I/