Yes, you can say something like...
"She just washed the dishes"
...but what exactly you mean is entirely context-dependent - you could continue with either of...
1: ...but she didn't dry them and put them away.
2: ...a few minutes ago.
Outside of context there's not normally any way to establish whether just means only a little while ago or only that and nothing else.
Regarding OP's secondary question about the meaning being affected by word position...
"She washed just the dishes"
...could only mean only/nothing but the dishes (she didn't wash the pots and pans). But so far as I can see this doesn't lead us to any useful general principle, given that...
3: "I just found out this morning."
4: "I found out just this morning."
...effectively mean exactly the same in most contexts (contrary to what might have been expected, I didn't know [whatever] until this morning). But there is a case for saying #3 is more suitable when emphasising that [whatever] is some piece of trivial information you happened to come across, and #4 fits better when you want to emphasise how recently you discovered whatever it was.
To summarise: for any given context, the position of "just" might affect meaning (subtly or decisively), but it's not obvious to me this leads to any useful general principles. In essence, "context is everything".