The difference between must and should doesn’t match with a difference of words in French but with a difference of moods. For must you have to use the indicative mood for either devoir or falloir (je dois, il faut...), for should  you have to use the conditional mood (je devrais, il faudrait).

In French, the vast majority of the time, “devoir” is similar to “ought to”, but in an informal context, it is understood as “should”. It is mainly used as Su.Ki. said, but not always. “Tu ne dois pas montrer du doigt!” You should not point! Here, “should” expresses an opinion, not a suggestion, which explains why “dois” is in the indicative mood, not the conditional one (devrais). If you really want to emphasise the conditional mood, especially in a formal context, “devoir” is normally understood as “ought to”. “Ça devrait fonctionner.” It ought to work. Here, the probability or the expectation is expressed. (“should” is mainly used in this case) “Il devrait être là pour sa fille.” He ought to be there for his daughter. Here, a moral obligation is expressed. (“ought to” is normally used in this case) “Must” = indicative of the verb “devoir”. (speaker’s feelings) “Je dois le dire.” (present) I must say it. “Je devrai le dire.” (future) Here, “will have to” is used in English. “J’ai dû le dire/Je devais le dire.” (passé-composé/imperfect) Here, “had to” is used in English. “Have to” = indicative of the verb “falloir”. (obligation due to external circumstances) Il faut le dire. It has to be said. “Falloir” is only used with the third person singular. To use the other persons in the indicative present, the conjugation of “avoir” in the “passé-composé” and the preposition “à” are used. “J’ai à acheter ce livre (pour telle ou telle raison).” I have to buy this book (for such and such reason). For the future tense, “avoir” is conjugated in the future tense, and “à” is still used. “J’aurai à le dire.” I will have to say it. For the past tense, “avoir” conjugated in the pluperfect and “à” are used. “J’avais à le dire.” I had to say it. I hope that helps.

For 'should', you use the conditional of 'devoir' : Je devrais le faire = I should do it