White leatherwork per the decree of organization of 15 floréeal An 11, and giberne the same as army foot artillery : so, crossed cannons.
Complete detailed regulation :
link
Summary :
Habit et veste de drap bleu, revers, parements et doublure de l'habit, de même couleur, bordés d'un liseré écarlate, avec un collet rouge montant, une pate rouge en long sur le parement de la manche ; le parement sera fendu et ataché par trois petits boutons, la culote sera de tricot bleu ; les boutons seront jaunes ornés de deux canons en sautoir avec une ancre transversale, le tout entouré d'une légende désignant le numéro du régiment par les mots premier ou second régiment. Le chapeau (bicorne) sera uni, bordé d'un galon de laine noire avec une ganse jaune et un petit bouton uniforme, cocarde nationale, pompon rouge. Les hommes reçoivent pour les travaux de force et corvées de port, ainsi que pour toutes les manœuvres d'artillerie et d'infanterie, un paletot de drap bleu et un pantalon de toile.
Les épauletes et distinctions des ofciers et sous-ofciers sont les mêmes que celle de l'artillerie de terre à pied. Les 1er canonniers porteront deux épauletes en laine rouge avec la frange de la même couleur, les seconds porteront deux épauletes rouges mais sans franges, Les aspirants-canonniers auront les deux épauletes en drap bleu liseré rouge. Les maîtres-canonniers auront un galon d'or large de sept millimètres (trois lignes) autour du collet de l'habit, les seconds maîtres le galon en laine aurore. Les sous- ofciers et canonniers de 1re classe ont le privilège du port du sabre. Apparaissent les chevrons d'ancienneté de laine rouge sur le bras gauche : un chevron pour 10 ans de service, 2 de 10 à 20 ans, 3 au-delà de 20 ans.
Other than adoption of the shako per order of 1806, I know of no other formal change to their uniforme or equipment. The change to shako may not have been 100% completed by 1813, or the chapeau was retained for walking out dress.
See : PDF link
However, if you google search images for "plaque de giberne" "artillerie de marine" "1er empire", you will see lots of items so offered at auction which have varying designs. Errors of the "expertes" ? Or real variations ? I do not know.
It was increasinly fashionable for officers, head-of-column members and maybe adjudant sous-officers to wear black leatherwork on campaign in 1813-1815. Also, any shortage at the home ports of the artillerie de marine of white leatherwork might have been filled by black leatherwork taken from inventory intended for sailors. But I know of no record of this happeming.
It was permitted per regulation to shift epaulettes to great coats when worn without the habit, but not to sew on rank stripes or seniority chevrons. The epaulettes would denote rank. Again, that said, shifting the sew-on insignia did occur in Spain, assumedly when the sous-officer's habit was no longer usable. The artillerie de marine entered 1813 with good quality, new-ish, habits. Unless ordered to toss them away, they likely left the sew-on insignia on the habits. On the other hand, Rousselot illustrates a sergent with his rank stripe on his great coat :
The illustration of the fanion de compagnie (above) is interesting. I would have expected the guide de droit (sous-officier de remplacement, typically the 1er sergent) or guide de gauche (sergent d'encadrement, typically the 2e sergent) to "jalloner" a compagnie. The illustration is of a 2e cannonier. While it is a completely plausible fanion design, I would love to know the artist's source (if any).
"When on campaign where their cartridge box covers white, like the line artillery and infantry, or were they black to match their belting?"
I would vote "white", if they even had them at all. The white was typically clothe with maybe a little linseed oil. The black was typically tarred light canvas. Neither would be sufficient on board to really protect the leather from salt spray. More likely the cover would just trap moisture against the leather, rotting it quicker in a marine enivoronment. Careful hot greasing or waxing, which could be done in a port garrison or on-board, would work better. The sailor's smooth black leatherwork was, in general, easier to hot wax/grease, as opposed to painted/powdered buffle blanche