How should I use "on" after the verbs to get the meaning of "continuously"?

"He/she goes on about" something, or "runs on" about something, is a way of complaining that someone never stops talking about a particular topic.

It is certainly not polite and casts ridicule.

But there are many other ways in which "on" can be used to express continuity - e.g.:

"It started to rain but the teams played on" means they didn't stop playing.

"The Titanic was sinking but the band played on".

"After the meeting many people stayed on for more than an hour".

"Despite their losses the battalion fought on".

The word on has an enormous range of uses. It is a preposition. Merriam Webster treats the usage that interests you as an adverb, as indeed it must be. This is what it says:-

2a: forward or at an advanced point in space or time

//went on home

2b: in continuance or succession

// rambled on;

//and so on

With some verbs you can just add on after the verb. So you can soldier on or speak on and on, walk on. But this is not always a standard usage, and it does not work in the case of the verb to try. Why? Unfortunately, on has been bagged by a different usage. We try on a pair of shoes or a sweater. Or, sometimes, we *try on *an excuse or made-up story, in which case someone might say to us sceptically, "you're just trying it on".

There are two verbs to which on can be added, and so, with the addition of the -ing participle, indicate continuity or persistence in the action. They are the verbs keep and go.

So you can keep on or go on running, shouting, waving, .... or trying. Equally, though, you can just keep trying.