\version "2.22.2"

#(define (integer-or-false? x)
   (or (not x) (integer? x)))

alt =
#(define-music-function (alt_up alt_down mus)
   (integer-or-false? (integer-or-false? #f) ly:music?)
   #{
     \tweak stencil
     #(grob-transformer 'stencil
                        (lambda (grob orig)
                          (if alt_up
                              (let* ((mu (markup #:fontsize -4 #:center-align
                                                 (alteration->text-accidental-markup (/ alt_up 2))))
                                     (altstc (grob-interpret-markup grob mu))
                                     (arttype (ly:event-property (ly:grob-property grob 'cause) 'articulation-type)))
                                (if (equal? arttype "trill")
                                    (set! altstc (ly:stencil-translate altstc '(0.7 . 0))))
                                (set! orig
                                      (ly:stencil-combine-at-edge orig Y UP altstc
                                                                  (if (equal? arttype "trill")
                                                                      -0.3 0.2)))))
                          (if alt_down
                              (let* ((mu (markup #:fontsize -4 #:center-align
                                                 (alteration->text-accidental-markup (/ alt_down 2))))
                                     (altstc (grob-interpret-markup grob mu))
                                     (arttype (ly:event-property (ly:grob-property grob 'cause) 'articulation-type)))
                                (set! orig
                                      (ly:stencil-combine-at-edge orig Y DOWN altstc 0.2))))
                          orig))
     #mus
                                  
   #})

\score {
  \relative c'' {
    \key c\minor
    c4-\alt ##f #0 ^\mordent
    c4-\alt #-1 ^\trill
    c4-\alt #0 #-1 ^\turn
  }
}

<<
  \new Staff \with { instrumentName = "Doc" }
  \relative c''' {
    a2(
    
    \once \override TextScript.script-priority = #-100
    a2-\tweak avoid-slur #'inside ^\prall^\markup { \sharp }
  
    \once \override Script.script-priority = #-100
    a2)-\tweak avoid-slur #'inside ^\prall^\markup { \sharp }
  }
  \new Staff \with { instrumentName = \markup\center-column { "My" "method" } }
  \relative c''' {
    a2(
    a2-\tweak avoid-slur #'inside -\alt ##f #1 ^\prall
    a2)-\tweak avoid-slur #'inside \alt #1 ^\prall
  }
>>

<<
  \new Staff \with { instrumentName = "Doc" }
  \relative c''' {
    \once \override TextScript.script-priority = #-100
    a2^\prall^\markup { \sharp }^\markup\italic"dolce"
  
    \once \override Script.script-priority = #-100
    a2^\prall^\markup { \sharp }
  }
  \new Staff \with { instrumentName = \markup\center-column { "My" "method" } }
  \relative c''' {
    a2-\tweak avoid-slur #'inside -\alt ##f #1 ^\prall ^\markup\italic"dolce"
    a2-\tweak avoid-slur #'inside \alt #1 ^\prall
  }
>>