You should rinse the eyes after cleaning but more to get rid of what you have just removed from the lid as opposed to the saltiness of the water.
The path you've got trapped in (walk in, get treated, discharge, flare-up, repeat) is going to make continuity of care very difficult.
I'd strongly consider going through the GP, therefore getting a named consultant for some consistency and preventative care.
As to eye drops vs ointment, it's generally considered you need both in tandem to get the best effect.
Drops during the day, ointment at night. For at least a month.
Although it's worth remembering that it is the lid hygiene that will get rid of most of the lid bacteria.
I'm somewhat torn about being any more explicit, I don't want to put you in the position of defending what a random mumsnetter said online.
So, we'll try this:
A few years ago Moorfields (the largest Eye Hospital in the UK) published the guidelines for managing this sort of issue (albeit slightly at the more severe end of the spectrum) in this paper.
The salient bits of treatment are repeated below.
"The children were treated with a regimen similar to that used in adult BKC. Daily lid hygiene was directed at removing debris and crusts along the lid margin, using cotton wool tipped swab sticks with diluted bicarbonate solution, diluted baby shampoo, or warm water.
The child?s attendant was advised that the lid hygiene routine had to be maintained for an indefinite period.
Patients with mild or moderate disease were prescribed a short course of topical broad spectrum antibiotics, usually chloramphenicol.
Drops were instilled four times daily and ointment applied to the lid margins at night for 1 month, after which they were instructed to use only the ointment at night for a further 2?3 months while maintaining a daily routine of lid hygiene. These patients were examined at 1, 2, 3, and 6 months following presentation, and then at 3 monthly intervals until they were stable using daily lid cleaning alone."
At least this way if you want to become the patient with the grande print-out (as opposed to the traditional patient with the petite note) you have something a bit more concrete than a webchat.
Sorry for length and hope that is of interest.