I suppose the problem is in discerning short ones -- long ones are fairly obvious.
Actually, I can easily tell apart both short and long ones.
Göre and
gòre sound like quite distinct words to me, even when the
e stays short.

The problem is that although I can easily register the difference, I don't actually feel the pitch going up or down; I just feel some abstract, inexplicable and unanalyzable difference. If I had been (hypothetically) taught about the classification of accents by someone who consistently used the opposite terminology (rising for falling and vice versa), I could honestly learn this material without ever noticing that there is anything wrong with it. I really feel nothing "falling" about the pitch of the first syllable even in, say,
majka, even though I clearly feel it as drastically different from the one in, say,
glava. (Once I actually took an online test of pitch perception, and I did very poorly.)
Thus the problem for me is not to feel the difference, but to remember the correct name for each category! Since my ear is not telling me what should be naturally called "rising" and what "falling", I have to memorize an example from each category, and then match the word with those. And since I don't discuss these matters too often these days, I keep mixing them up...

Still, thanks a lot for the advice below; some of it could really make things easier for me in the future.
If I may offer a couple of tips that work for me (which don't necessarily mean they will work for you

):
1) Use the fact that rising accents can't occur in monosyllabic words: pronounce the word, but "swallow" the second and later syllables. If the remainder is pronouncable, you got a falling one. e.g:
nizak has
niz-; that niz- "rhymes" with "niz vodu", i.e. it's pronouncable as single word, i.e. it's falling.
But e.g.
visok has
vis-. That one "wants" a second syllable, and doesn't "rhyme" with
ris. It's rising.
Yes, this is definitely a very good mnemonic. Although in some places in Bosnia, a single long rising "a" is used as a word (a question marker of sorts).
2) "Sing" the word in your mind (or aloud, but be prepared for strange looks if you get spotted ), but mutter "mhms" rather than real syllables. Pay attention to the pitch difference, and exaggerate it if necessary. If the first syllable is notably higher than the second, it's falling. If they're about the same, it's rising. Practice with nizak and visok.
With this part, I have no problems. As soon as I have sample words from each category, I can easily match them mentally with any others. And my tone-deafness unfortunately holds for comparisons between syllables too.
P.S. A case could be made for splitting the thread, starting from
this post.
Yes, it should be split; we've now drifted apart from the original topic completely. (But not aimlessly, although I'm sure that most speakers of European languages would consider the idea of conveying definiteness by tones as supremely bizarre.

)
"Taj vedri momak" can be analyzed as "taj ved [ri] momak" (rule 1) thus it's falling, i.e. vëdrī (I pronounce length here - do you?)
Yes, I pronounce it (in fact, quite prominently).
Also, there is another interesting thing I've noticed about my perception of accents: my brain lumps together the accent and the post-accentual length into a single piece of information! In the famous example
göre gòre görē gòrē, I clearly perceive each word as different from the other three -- but if you asked me to pair them up so that words in each pair have the same accent on
o and differ only in the length of
e, I'd have a hard time doing it. However, now I realize that the above method (1) can help a lot with this.