Both "to put", and "to get" do indeed seem to have old Germanic roots from their etymology.
to put
...probably from a Germanic stem that also produced Danish putte "to put," Swedish dialectal putta; Middle Dutch pote "scion, plant," Dutch poten "to plant," Old Norse pota "to poke." Etymology Online
to get
...from Proto-Germanic getan [...]. Old English, as well as Dutch and Frisian, had the root only in compounds (e.g. begietan "to beget," see beget; forgietan "to forget," see forget) [...] Etymology Online
In contemporary German not much of the Germanic roots is preserved. However there are some remnants, at least in dialects from the northern regions, or in compounds.
Here are some examples I found:
paten/poten: (to plant)
verb. pflänzlinge setzen: "als ob er nichts weiter gethan hätte als weiden paten". Bode Montaigne 3, 70. ein nd. wort, abgeleitet von mnd. pate, pote, potte, nd. pate, setzling, pôte, pfropfreis [...] Grimm
vergessen: (to forget)
Es handelt sich um präfigierte Verben zu einem Simplex, das in ahd. geʒʒan ‘erlangen’ (8. Jh.), anord. geta ‘schaffen, erreichen, erzeugen, lernen, nennen, vermuten’ (woraus mengl. geten, engl. to get ‘erhalten, bekommen’) DWDS