How are you powering the GPS module? What voltage, filtering etc? It doesn’t sound like the lack of level converter is causing your problem, and the antenna looks like a suitable match…
The GPS-00177 antenna works as good as any. And 5V parts will usually accept 3.3V levels with no problem (at least I’ve never seen a case where they wouldn’t) so I highly doubt you need a level converter which is more to protect a 3.3V input from a 5V part if the 3.3V input isn’t 5V tolerant.
At first, by ‘touch’, I thought you meant when the metal antenna part or connector was touched (as in connected or shorted) to the GPS but I think what you meant was it works when you touch a metal part of the antenna or the connector with your finger. If so, it sounds like you may have an RFI problem (RF interferance from some wireless device, a PC or anything else close by with a processor or high freq clock signal) and that touching the antenna helps reduce the level of the RFI or at least helps prevent it from reaching the antenna. If you have a scope you should also check the output of your 3.3V reulator to ensure it’s not oscillating or simply make sure there’s an input and output capacitor close to the regulator to ensure it’s stable. As a note, I’ve never had a problem with a Copernicus, but Lassen iQ GPS receivers will not work anywhere within about 15ft of my main PC due to RFI.
30-35 sec’s is typical for a fix, but I have no idea as to the magic, other than it could be RFI. After my Lassen iQ experiance I’m always on the lookout for RFI problems as I use GPS to track and recover high altitude balloon payloads and can’t afford to have a problem with a camera or some other equipment in the payload box generating RFI and affecting the GPS. Most active antennas include a filter to help reject frequencies close to the GPS signal freq. and all I can think of is perhapes the Globalsat Antenna has a better filter or at least a filter with a slightly different response that helps reduce some particular local RFI frequency. Do you have a different computer you could try using in case it’s RFI from the computer?