Wow, such varied advice, some of it is bound to be wrong. I was holding back waiting for the dust to settle. Until I retired three weeks ago :-) I owned and operated a business that did furniture repair and refinishing, including kitchen cabinet refurbishing.
First thing you need to do is to determine what is on there now. Could be a vinyl coating, RTF (thermo-foil), lacquer, pre-cat lacquer, UV cured, conversion finish, or waterborne finish. It is almost certainly not "polyurethane" that you can buy at Home Depot. Most people think all clear finishes must be poly because that's all they see at the paint store or big box. In 11 years of this work, I never saw a polyurethane finish on a factory-finished piece. It's simply not a production finish. Though there are some two-part polys, they are very specialized and I never encountered one. If it's a plastic or conversion finish, cleaning may be all you can do because finishes won't stick to that stuff.
First, I would suggest just a good cleaning, both with a detergent like Dawn and water to remove the water-soluble stuff, and with mineral spirits to remove the non-water soluble stuff. In my industry, these are known as "polar" and "non-polar" solvents. See below for the link that explains how to clean.
The fact that you have used Pledge is unfortunate. Its silicone oil will upset the surface tension on almost all finishes and cause them to fish-eye (flow out in small little craters). Shellac (Bullseye is one brand in aerosol, and Sealcoat is a canned version that's dewaxed), are both by the same company, Zinsser. This is one of the few products that will not fish eye. In addition, shellac will stick to most any finish, and most finishes will stick to it if it's de-waxed. There are fish-eye preventer additives, but you'd need to have a spray system set up. Go with the aerosol shellac. Pledge, in my book is evil and I've come to hate it. I'll stop there.
Other comments:
- I have found Murphy's oil soap much less effective than Dawn. I've worked on some old pieces that the customer had cleaned six times with it and using my products, I'd get about 1 square foot done before the cloth was so black that I'd need to get a new one.
- I have found Howard's Restor-a-Finish little more than a furniture polish with a little denatured alcohol and a little color added. It looks good for a few weeks, then the product seems to have evaporated away and it looks like it did before.
- Lemon Oils are typically just light mineral oil with a dash of yellow dye and lemon scent. Adds a little sheen and slowly evaporates.
- There are various versions of Thompson's Water Seal. The oil-based ones seem to be a lot of thinner ("light aromatic hydrocarbons", naphtha, mineral spirits, stoddard solvent, or description choice of manufacturer) and a splash of wax. Rather than go this route, just buy some Johnson's Paste Wax, Minwax, Liberon tinted waxes, etc., if you are so inclined. Finishing guru, Bob Flexner says, "A wax finish is about the closest thing there is to no finish at all." Just because water is beading up, that's surface tension and is not preventing the water from penetrating.
- Briwax contains toluene (aka tolulol), that is a fairly strong solvent and can attack some finishes, namely lacquer. It's meant as a restoration wax. I've seen it damage some finishes. Briwax II uses mineral spirits that is safer. And no, mineral spirits will not damage finishes. I've seen it pull some color on one rustic Asian pieces that were finished with a concoction of mineral spirits, tar (Gilsonite) and a bit of linseed oil. Again, really rare.
- Best treatise on the subject
Saving the finish