'As best you can'... does'nt sound good enough. You must get rid of nearly all the old sealant.
When I do it, I usualy use a stanley type blade as a scraper, so as to get right down to the tile surface.
When you are happy with the preparation remove all dust with a vacum.
If you want a proffesional even finish get some of these Fugi tools, they are brilliant.
You just overfill the area between the top of the bath and the bottom of the tile, and scrape off the excess with the Fugi ... I cant be without them for finishing worktops, basins and baths.
When I've siliconed around a bath, the owners often say I must be really skilled to make such a good finish with the silicone ... I never disagree ... but I dont tell them about my little secret ... watch the video.
http://www.byretech.c...5aSzaMCFQ1t4wod0l5ewQ