Excellent work, Jack, and welcome aboard!

> halda með landinu fram
> they keep following the land in front

'fram' here is an adverb indicating motion, the direction they're
going in: they carry on forwards along the coast.

> settu enn stafn við því landi
> they set the prow in to the land

'enn' "again", rather than 'inn' "in".
'við' "against".

> En veður óx í hönd
> But storms grew in that place?

"But the wind grew higher and higher." (Idiomatic.)

> when he had nothing to say about those lands
> er hann hafði ekki að segja af þeim löndum

'er' can mean "when", but can also introduce a clause that gives the
reason for something, or an explanation of the circumstances, so
annother way of translating it might be: "as he had nothing to say
about those lands", or something like that.

> um sumarið eftir
> roughly the following summer

Leave out "roughly". The 'um', in this sentence, is just part of the
idiom: 'um sumariðp eftir' "the following summer".

> og keypti skip að honum
> and bought a ship for himself?

"and bought a ship from him" (See Zoega 'at' definition number IV.8.)
If he'd been buying the ship from himself(!), it would need the
reflexive pronoun 'sér' in place of 'honum' (which can't refer back to
the subject of the sentence).

> réð til háseta
> and decided on oarsmen

"hired oarsmen for [a voyage]" (See Zoega 'ráða' 7; 'til' 4).

> Leifur bað föður sinn Eirík að hann mundi enn fyrir vera förinni.
> Leifr bade his father Eirik to still come on the voyage?

"[...] that he should be in command of the voyage again." (As he had
been in command of the voyage to Greenland?)

But well done! You seem to have a good understanding of the grammar.