2 older dwarf fruiting pear trees produce new growth from their bases. Is there any advantage at all in leaving one of these new growths particularly on a neglected damaged tree or must all fruit trees be single trunk/stem?
Since your trees are dwarf, they must be grafted on a dwarfing rootstock which, for pears, is usually not even pear – it is quince! So the suckers are quince and are unless you want a quince tree, I would buy a new one, see http://walcotnursery.co.uk/ for a good range of varieties, organically grown and at reasonable prices. I have found all their trees to grow really well.