The verb have [meaning 'possess; not the auxiliary]:
We make questions and negatives with have in two ways:
normally we use do/does or did for questions :
Do you have plenty of time?
Does she have enough money?
Did they have any useful advice?
I don’t have much time.
She doesn’t have any money.
They didn’t have any advice to offer.
… but we can make questions by putting have, has or had in front of the subject, optionally with 'got' after the subject. Using 'Have you' rather than 'Have you got' with many noun groups, especially short ones, often sounds at least slightly unidiomatic, old-fashioned, or po-faced: Have you got a cold? Have you got some sugar? // (?)Have you a cold? (?)Have you some sugar? The particular noun group seem to be important; 'Have you an appointment?' sounds far more acceptable that 'Have you a cough?'. 'Have you toothache?' and 'Have you ankylosing spondulitis?' sound bizarre.
But note that the have's in 'Have you an appointment?' and 'Have you got an appointment?' are different, being a main verb and an auxiliary respectively.
Have you plenty of time? / Have you got plenty of time? (either sounds fine here)
Had they any useful advice? / Had they got any useful advice? (the first sounds at least as good!?!)
… and we can make negatives by putting n’t, n’t got or not got after have, has or had:
We haven’t much time. // We haven’t got much time. // We have not got much time. // (?)We have not much time.
She hadn’t any money. // She hadn’t got any money. // She had not got any money. // (*/?)She had not any money.
He hasn’t a sister called Liz, has he? // He hasn’t got a sister called Liz, has he? // He has not...
British Council: English Grammar [tweaked]
Summing up: I'd say that there has been a general move towards the use of the usages you say are new, and that the 'older style' you mention ('Have you a cough?') is perceived as dated, though as Barrie points out, 'I have not a car' was not in common use even 50 years ago.