First, you need to use the present participle in these cases. "Without" is a preposition in these sentences and so requires a noun or pronoun as its object. The prsent participle of a verb can be used as a noun (meaning the act or state represented by the verb) and is then called a gerund. The perfect participle cannot be used as a noun. So
I cannot live without hugging you
is the correct form.
Second, modals do not have a future tense, and some (e.g, "must") do not have a past tense. You have to work around that.
I shall not be able to live without hugging you
is grammatical but ambiguous between
I shall not be able to live without hugging you now
I shall not be able to live without hugging you in the future.
Third, putting this thought into the simple past sounds strange to me.
I could not live without hugging you
is grammatical, but seems more like a conditional statement rather than a past statement. To make it clearly about the past, try
I could not have continued living without having hugged you.