"debeo" is better translated here as "I ought to" or "I am obliged" or "I am bound to" rather than "should"
Agreed; it's both better reflective of the meaning as well as giving a more literal translation with visitare as an infinite: "I am obligated to visit (the) mother".
Your suggestion and thenino85's are good. I think the shortest, literal rendering is "I must." The conditional form of debeo is probably better suited to "should."
Is it grammatically wrong if I rearrange the verbs order? matrem visitare debeo vs matrem debeo visitare. What I learn from another romance language is that the verb order should be the other way around, in Italian - devo visitare.