It means
I've almost but not quite given up dating ...
The All but part suggests the speaker has done everything.
But the expression is used when the speaker can't think of anything further to do, so it might not be literally everything.
Q1. How to distinguish between past participles used as adjectives and past participles that have a passive meaning?
Past participles that have a passive meaning have an explicit or implicit agent.
- She is admired by everyone who knows her (explicit).
- Your help is appreciated (implicit, by me).
- The price of petrol has been reduced (implicit, by the oil company).
Such past participles are typically not modified by very, but by very much or an alternative adverb:
- She is greatly admired. (?She is very admired.)
- Your help is very much appreciated. (?Your help is very appreciated.)
- The price of petrol has been significantly reduced. (?The price of petrol has been very reduced.)
Past participles used as adjectives very often describe mental or emotional states, and therefore have a person or animal as their subject. There is no explicit agent, and often not even an implicit one. Such past participles are typically modified by very, not by very much.
- I'm very bored. (?I'm very much bored.)
- John's been very depressed for several days. (?John's been very much depressed for several days.)
- She looked very disappointed. (?She looked very much disappointed.)
Q2. Which adjectives to use with very and which adjectives to use with very much.
As noted in the section above, past participle adjectives that describe mental states are generally modified by very, not very much. Most other adjectives are also modified by very alone. However, there are some common exceptions. For example, adjectives that describe extreme qualities are not usually modified by either very or very much (?very enormous, ?very much wonderful). There is another group of adjectives that needs a different modifier than very (fast asleep, wide awake, far apart, well known, etc.)
Q3. "This looks very like what we had at our shooting party in November." Is this sentence is really correct?
A short answer: yes, This nGram shows is very much like to be more common currently than is very like, but before 1940 the reverse was the case.
In answer to your supplemental question (How to determine which form to use?), my recommendation would be to invest in a good grammar of English usage* to learn about general patterns, and run an nGram as above on specific instances (or do a simple Google search on the two phrases: for example "is very like" gets 411,000 hits, while "is very much like" gets 36 million - so it is clear which is the preferred form).
*The two books I consulted in preparing this answer were Swan's Practical English Usage and Collins Cobuild English Usage.
Best Answer
The phrase the way in both instance means the manner.
In the first instance, it is an adverbial phrase modifying the adjective wonderful (which is a predicate adjective modifying she). It could also be expressed
The way and what follows explains why she is wonderful.
The second instance is a direct object
The way and what follows is what he loves. There is an implied that following way beginning an adjectival clause defining way.