Is it only the higgs field that stops me from accelerating up to the speed of light or is there other restriction concerning the increase in mass? In other words if I had a Higgs field shield on my spacecraft could I get close to the speed of light without an increase in mass?
Courtesy of the question Bound State of Only Massless Particles? Follows a Time-Like Trajectory? the answer to your question is no.
If you take the example of a glueball formed from two gluons, although the gluons are massless the glueball has a rest mass. In his answer to the above question Ben Crowell argues that the glueball must move on a timelike trajectory i.e. slower than the speed of light. So we have an example of a bound state of massless particles that moves slower than $c$.
So even though removing the Higgs contribution to the mass would render the elementary particles massless, QCD bound states would remain massive and travel on timelike trajectories.