I think that too much is made of these terms offshore or coastal cruisers. While there can be major distictions between how a specific boat is intended to be used, when you talk about extended cruising in areas that are somewhat breezy in nature, then you are better off in a boat that has certain kinds of attributes, (such as robust construction, good deck hardware, good sails and reefing gear, seaberths, plenty of storage and tankage, small but operable portlights, and a comparatively small cockpit with large drains.)
If you are an experienced sailor with good boat repairing skills, you can by with a boat that compromises on some of these characteristics and can upgrade the boat to over come any serious deficiencies.
When you talk about the Carribean you are talking about a large region with a wide varieties of sailing conditions. Areas like the Vigin Islands offer a wide range of marine services within a day or so sail of anywhere. The conditions are quite breezy but rarely more than manageable and when things get dicey there are plenty of places to duck in and wait things out. Other areas of the Carribean you are several days away from safe ports of refuge and conditions can be less predictable.
People have cruised these areas in allkinds of boats. It can be done with some skill and some luck. That said, this all comes down to risk management. How much are you willing to risk? If you end up buying an unsuitable boat, the chances are greater that something catastrophic will happen. You are clearly on a tight budget. That catastrophy may not be that you sink your boat or loose your
rig. It may simply be that you loose some part of the boat that you cannot fix with spares onboard in some remote corner of the world and cannot affort to buy the part and have it shipped to where you are and are forced to leave your boat in a hurricane zone and return to the States to refresh the kitty. While this can happen on any boat, it is less like to happen on a more robustly constructed boat and one designed for the kind of proplonged exposure to the kind of conditions encountered in these breezier corners of the world.
Given your budget, probably the best way to go may be to buy an older, tougher boat from the 1960''s or early 1970''s that someone has spent time restoring, and then spend some more time going over the boat and carefully upgrading those few items that might have been missed by the previous owner.
Jeff