I know that there is a filedialog command in lite-c but this does not work well in fullscreen mode (it crashes the game on some computers). Could there be one which works in fullscreen too?

Its not crashing the game, Its freezing the window as long as it is opened. In fullscreen its sometimes not in the front. However, there are 2 ways:
1. Cut the borders of the engine window and size it on current resolution but dont turn on fullscreen.
2. Make a dialog yorself with panels and buttons.

you can use Texts and Panels to create something similar to a file dialog
the problem is that is really much work and also not as good as the original ones

here are some functions which may help:
txt_for_dir(text,STRING* name)
txt_sort(TEXT*);
txt_load(TEXT*,STRING* filename)
panel->event
pan_setdigits
pan_setbutton

also oliver2s has created a fully functional file dialog for his IceX3:
Forum Thread

MQ32's hint are really fine.
I also created my own file save/load panel in lite-c, bacause in a 3D environment it is usually necessary to fit its style to other panels/menus.
I use:
- a slider and 2 buttons for scrolling (slider position handling requires a loop),
- fixed amount of strings (an array) dislpaying file names read from a text created by txt_for_dir, due to slider position and file name quantity
- buttons under file name strings, for easy highlighting selection and mouse touch of filenames
- ok and cancel buttons
- some title text strings
- a text object displaying actual filename or for typing custom filenames - as I remember it requires a loop to update filename data without hitting enter
- some button bitmaps in Paint