Don't you usually pronounce the final consonants in your language? If you don't, then your first language is interfering with your second language.
Actually I don't think your interpretation is correct when you say you can't make your speech flow smoothly if you pronounce the final consonants. On the contrary, you can only make your speech flow smoothly if you do pronounce the final consonants, at least if the next word begins with a vowel sound..
What can be done to remedy your problem? Other teachers may offer different solutions and it would be interesting to have their advice because I usually seem to have at least one Asian student in each of my classes who has the same problem.
My own advice would be for you to take a short text and highlight or underline all the final consonants. Then carefully and slowly record the text on a tape recorder, making sure you pronounce clearly all the highlighted or underlined consonants.
Later, listen to your recording, pausing regularly, comparing it carefully with the text. If you have forgotten to pronounce a consonant, do the recording again until you get it 100% right.
Your highlighted text should look like this (taken from the notice printed above):
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Practice makes perfect. Once you have done this sort of activity a few times, you should be able to overcome your problem.