I'm wondering if both of these are correct ways of saying 'I have to', and if so, whether there's any difference in what is implied:
Je dois acheter (un parapluie) VS J'ai acheter (un parapluie)
The first form is correct but the second one is not grammatical and would also be confused with j'ai acheté un parapluie (I bought an umbrella) in spoken French.
You might say:
J'ai à acheter un parapluie. (very rare, might be perceived as an anglicism)
Je n'ai pas à acheter un/de parapluie. (common)
J'ai un parapluie à acheter.
As @jlliagre said the second sentence is not correct.
If you want to say "I have to" you can also say
Il faut que j'achète un parapluie