Are "advise somebody doing something" and "suggest somebody doing something" grammatically correct?
I suggested us (rather than "we") going in my car.
I advised us (rather than "we") going in my car.
In the case of "suggest", the two options that come to mind are:
I suggested going in my car.
I suggested (that) we go in my car.
You cannot say directly "to suggest someone something", nor can you say "to suggest (to) do something".
See further explanation on English Teacher Melanie and more examples of the correct usage of this verb on Oxford Learner's Dictionaries.
Using "advise" doesn't sound right in this context. If there were a third person saying it, the sentence could be:
He advised us to go in my car.
More correct usage of "advise" on Oxford Learner's Dictionaries.
I suggested us/we going in my car.
I advised us/we going in my car.
Both the sentences are grammatically incorrect. You can rephrase these sentences with the use of the conjunction "that" as follows:
I suggested that we go in my car.
I advised that we go in my car.
You can use "should" before the verb in these sentences in British English.