Depending on context, this could also mean something like, "Let's try again"
Correct, this is a very common way to express let's in Swedish. :)
Is there a literal way to say "let us" even if it's not commonly used?
Låt oss. Sounds very very old fashioned and uptight.
Adjusts monocle hmm-hmm yeeeess, forsooth, let us retire to the parlor whereat I can show you my collection of antique soaps and ill gained fossils
In Dutch there's a word that sounds a lot alike försöker: verzoeken. It means to ask someone to do something. Is there a Swedish word for that? Or can försöker be used as well?
You'd say be in that case. Jag ber dig att försöka = I ask you to try.
Försöker translated to german means versuchen. Swedish is so similar and easy! :D
Could this also mean "We will try again."?
Yes, the Swedish present is often used to refer to the future, depending on context but it's a very likely interpretation here.
Okay, good luck this time!