tubes and current

I don't have the electrical/electronic chops to answer, maybe someone else will have to provide an answer but it seems the amplifier design,  power supply and transformer that will determine current.

I don't think "high current" determines what your speakers see. 

Speaker impedance (resistance) come into play.   Room size comes into play.

Your B&W's have a nominal impedance of 8 Ohms.   With a recommended amplifier power rating of 50 - 120W into 8 Ohms unclipped.

As long as you match the speakers impedance to the amplifier in use, so that the amp is not working overtime, you should be fine.  It's when we ignore the manufacturer's parameter's that we get into trouble.

You have a severe amp/speaker mismatch. Your amp is designed for a high sensitivity, benign impedance speaker. The B&W805 D3 is not even remotely close to being that type of speaker and better suited for a solid state amp that is stable with a low impedance load. Many years ago I tried a 2 way bookshelf speaker on a 300B SET amp I had. This speaker was not quite as demanding as the B&W with a 2db higher sensitivity and 25 watt recommended minimum power. The sound needed just what you're looking for, dynamics and more body. The proper amp/speaker match made a night and day difference in the sound of that speaker. Even though the B&W is better suited for solid state, it's possible to work with tubes, however you need to adhere to the recommended minimum of 50 watts. Some music can have dynamic peaks of 20db and more. So if you're using 1 watt, a 15db peak requires 32 watts, and a 21db peak would require 128 watts. Even with the EL34 amp you tried, I seriously doubt you were getting anywhere close to the performance that B&W is capable of.

Thanks for the suggestions so far.

Don't change the amp change your speakers

Your speakers and your amp are just about as polar opposite as one could get when trying to match an amp with a pair of speakers. You can read up on tube circuits, current delivery, current vs. voltage source amps, etc., etc., but the bottom line is you need to decide do you want to keep the amp or the speakers. If you decide to keep the speakers, you'll need to make a significant investment to get a tube amplifier that could drive them to their full potential. Or you could make a fraction of that investment and get a solid state amplifier that would give you all you need minus the tube "flavor". If you decide to keep the amp, look for speakers with efficiency/sensitivity of >96dB, full-range single driver or otherwise, and you'll be very happy with the results, assuming you like that sort of sound. Hope this helped.